An essay on probabilities : and on their application to life contingencies and insurance offices

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Professor of Mathematics in University Collage; and

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Secretary of the Royal Astronomical Society

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PRINTED FOR LONGMAN, BROWN, GREEN &

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PREFACE.

In order to explain the particular object of this Trea-
tise, it will be necessary to give a brief account of the
science on which it treats.

At the end of the seventeenth century, the theory of
probabilities was contained in a few isolated problems,
which had been solved by Pascal*, Huyghens, James
Bernoulli, and others. They consisted of questions re-
lating to the chances of different kinds of play, beyond
which it was then impossible to proceed: for the dif-
ficulty of a question of chances depending almost en-
tirely upon the number of combinations which may
arise, the actual and exact calculation of a result be-
comes exceedingly laborious when the possible cases are
numerous. A handful of dice, or even a single pack of
cards, may have its combinations exhausted by a mode-
rate degree of industry: but when a question involves
the chances of a thousand dice, or a thousand throws
with one die, though its correct principle of solution
would have been as clear to a mathematician of the six-
teenth century as if only half a dozen throws had been
considered ; yet the largeness of the numbers, and the

* Un probléme relatif aux jeux de hasard, proposé @ un austére janse-
niste par un homme du monde, a été l’origine du calcul des aegeag a Ge
oisson.

A

CMQGROYD

vi PREFACE.

consequent length and tediousness of the necessary
operations, would have formed as effectual a barrier to
the attainment of a result, as difficulty of principle, or
want of clear perception.

There was also another circumstance which stood in
the way of the first investigators, namely, the not hav-
ing considered, or, at least, not having discovered, the
method of reasoning from the happening of an event to
the probability of one or another cause. ‘The questions
treated in the third chapter of this work could not
therefore be attempted by them. Given an hypothesis
presenting the necessity of one or another out of a
certain, and not very large, number of consequences,
they could determine the chance that any given one or
other of those. consequences should arrive ; but given an
event as having happened, and which might have been
the consequence of either of several different causes, or
explicable by either of several different hypotheses,
they could not infer the probability with which the
happening of the event should cause the different hypo-
theses to be viewed. But, just as in natural philosophy
the selection of an hypothesis by means of observed
facts is always preliminary to any attempt at deductive
discovery ; so in the application of the notion of proba-
bility to the actual affairs of life, the process of reasoning
from observed events to their most probable antecedents
must go before the direct use of any such antecedent,
cause, hypothesis, or whatever it may be correctly
termed. These two obstacles, therefore, the mathema-
tical difficulty, and the want of an inverse method, pre-
vented the science from extending its views beyond
problems of that simple nature which games of chance
present. In the mean time, it was judged by its fruits;

PREFACE Vil

and that opinion of its character and tendency which is
not yet quite exploded, was fixed in the general mind.
Montmort, James Bernoulli, and perhaps others, had
made some slight attempts to overcome the mathema-
tical difficulty ; but De Moivre, one of the most pro-
found analysts of his day, was the first who made
decided progress in the removal of the necessity for
tedious operations. It was then very much the fashion,
and particularly in England, to publish results and con-:
ceal methods ; by which we are left without the know-
iedge of the steps which led De Moivre to several of his
most brilliant results. These however exist, and when
we look at the intricate analysis by which Laplace ob-
tained the same, we feel that we have lost some im-
portant links * in the chain of the history of discovery.
De Moivre, nevertheless, did not discover the inverse
method. This was first used by the Rev. T. Bayes, in
Phil. Trans. liii. 370.; and the author, though now
almost forgotten, deserves the most honourable remem-
brance from all who treat the history of this science.
Laplace, armed with the mathematical aid given by
De Moivre, Stirling, Euler, and others, and being in
possession of the inverse principle already mentioned,
succeeded both in the application of this theory to more
useful species of questions, and in so far reducing the dif-
ficulties of calculation that very complicated problems
may be put, as to method of solution, within the reach
of an ordinary arithmetician. His contribution to the
science was a general method (the analytical beauty and
power of which would alone be sufficient to give him a
high rank among mathematicians) for the solution of

wp The same may be said of several propositions given by Newton.

(|

Vill PREFACE. | }

all questions in the theory of chances which would
otherwise require large numbers of operations. The
instrument employed is a table (marked Table I. in the
Appendix to this work), upon the construction of which
the ultimate solution of every problem may be made to
depend.

To understand the demonstration of the method of
Laplace would require considerable mathematical know-~
ledge ; but the manner of using his results may be de-
scribed to a person who possesses no more than a common
acquaintance with decimal fractions. To reduce this
method to rules, by which such an arithmetician may
have the use of it, has been one of my primary objects
in writing this treatise. JI am not aware that such an
attempt has yet been made: if, therefore, the fourth, and
part of the fifth chapters of this work, should be found
difficult, let it be remembered that the attainment of such
results has hitherto been impossible, except to those who
have spent a large proportion of their lives in mathe-
matical studies. I shall not, in this place, make any
remark upon the utility of such knowledge. Those who
already admit that the theory of probabilities is a desir-
able study, must of course allow that persons who cannot
pay much attention to mathematics, are benefited by
the possession of rules which will enable them to obtain
at least the results of complicated problems ; and which
will, therefore, permit them to extend their inquiries
further than a few simple cases connected with gambling.
By those who do not make any such concession, it will
readily be seen, that the point in dispute may be argued
in a more appropriate place than with reference to the
question whether others, who hold a different opinion,

PREFACE. ix

should, or should not, be supplied with a certain arith-
metical method.

The first six chapters of this work (the fourth, and
part of the fifth exclusive) may be considered as a
treatise on the principles of the science, illustrated by
questions which do not require much numerical com-
putation. To this must be added the first appendix,
on the ultimate results of play. Omitting the first pages
of the latter, the discussion on the noted game of rouge et
noir will, with the problems in page 108. &c., serve to
show the real tendency of such diversion. I am informed
that this game is not played in England at any of the
clubs which are supposed to allow of gambling: but it
was permitted in the Parisian salons until the very
recent suppression of those establishments ; and the ac-
count given of it will show what has taken place in our
own day. The game of hazard is more used in this
country; but I have been prevented from giving it the

same consideration by the want of aclear account of the

manner in which it is played. Nothing can be more
unintelligible than the description given by the cele-
brated Hoyle.

The fourth chapter has been already alluded to: it
contains the method of using the tables at the end of
the work in the solution of complicated problems. The
seventh chapter, and the fourth appendix, contain the
application of the preceding principles to instruments of
observation in general.

The remainder of the work is devoted to the most
common application of this theory, the consideration of
life contingencies and pecuniary interests depending
upon them, together with the main principles of the
management of an insurance office. As this portion was

x PREFACE.

not written for the sake of the offices. but of those who
deal with them, I have confined myself to such points
as I considered most requisite to be generally known.
Common as life insurance has now become, the present
amount of capital so invested is trifling compared with
what will be the case when its principles are better un~
derstood ; provided always that the offices continue to act
with prudence until that time arrives. At present,
while the public has little except results to judge by, the
failure of an office would cause a panic, and perhaps re-
tard for half a century the growth of one of the most
useful consequences of human association : but the time
will come when knowledge of the subject will be so
diffused, that even such an event as that supposed, if it
could then happen, would not produce the same result.

There are, however, one or two things to which I
should call the attention of those whose profession it is
to calculate life contingencies : —

1. The notation for the expression of such contin-
gencies (pp. 197—204.). This notation was suggested
by that of Mr. Milne, from which it differs in what I
believe to be acloser representation of the analogies which
connect different species of contingencies. Thus, an
annuity to last a number of years certain does not differ
from a life annuity in any circumstance which requires
a difference of notation ; nor an insurance from an an-
nuity certain of one year deferred till a life drops.
Since writing the pages above referred to, I have learned
that I was not the first who considered an insurance in
that light. Some years ago the government granted
annuities for terms certain, ‘to ccmmence at the
death of an individual ; but refused to insure lives: the
consequence was, that, by a very obvious evasion, insur-

PREFACE, xi

ances were effected by buying annuities for one year
certain, to commence at the death of a person named.
This had the effect of putting an end to such annuities.

2, The form of the rule for computing the value of
fines, and its introduction into the method of calculating
the present value of a perpetual advowson (pp. 231. 236.
and Appendix the Second). It will be found that the
rule of every writer on the subject is palpably wrong
in principle, with the exception of that of Mr. Milne.

3. The rule for the valuation of uniformly increasing
or decreasing annuities, given in the fifth appendix. A
simple application of the differential calculus is made a
striking instance of the position, that the labour of a
person of competent knowledge is seldom lost. The
annuities given by Mr. Morgan and Mr. Milne, are for
every rate of interest, from three to eight per cent.;
and perhaps those gentlemen may have had some doubts
as to the necessity of inserting the two last rates. It
now appears, however, that, in consequence of the extent
to which their tables are carried, the values of increasing
or decreasing annuities, can be calculated with great
accuracy for three and four per cent., and with sufficient
nearness for five per cent. ; and with very little trouble,
compared with that which it must have cost Mr. Morgan
to calculate the table referred to in page xxviii. of the
Appendix.

The rules, in page xxix. of the Appendix, contain a
point which, as no demonstration is given, may cause
some difficulty. In turning an annuity or insurance
which cannot be extinguished during the life of the
party into one which can, a direction to add is given
which will at first sight, perhaps, be supposed to be a
mistake, and that subtract should be written instead. But

Xli PREFACE.

it must be remembered that an annuity of, say £3 a year,
diminishing by £1 every year, is equivalent, by the
first part of the rule, to an annuity of which the suc-
cessive payments are as follows:

£3, £2, £1, £0, £(—1), £(—2), £(—3), &e.

That is, the first part of the rule, when the annuity is
extinguished during the tabular life of the party, gives
the value of his interest upon the supposition that he
is to begin to pay as soon as he ceases to receive. If
then, this is not to be the case, the value of his interest
must be increased accordingly.

4, The method of the balance of annuities, or the
determination of complicated annuities by the addition
and substraction of simple ones. This has been done
before ; but it has not, to my knowledge, been carried to
the extent of making all the questions which commonly
occur deducible from the fundamental tables, without
the aid of any new series. It is desirable that the
beginner should be accustomed to deduction by reasoning,
without having recourse to the mechanism of algebra,
which, as a quaint editor of Euclid observed, “is the
paradise of the mind, where it may enjoy the fruits of
all its former labours, without the fatigue of thinking.”
Of no part of algebra is this more true, than of the
method by which complicated annuities are deduced
from simple ones, by the resolution of the series which’
represent them into the simpler series of which they are
composed. The education of an actuary does not neces-
sarily imply the study of geometry ; and such processes,
for instance, as those by which are found the values
of a contingent insurance or a temporary insurance
(pp. 222. 226.), will serve, as far as they go, to ac-

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PREFACE. Xili

custom him to make those efforts of mind, and to bear
that tension of thought, the necessity for which is the
distinction between a problem of geometry, and one of
ordinary algebra.

The considerations contained in this volume have, in
my opinion, a species of value which is not directly de-
rived from the use which may be made of them as an
aid to the solution of problems, whether pecuniary or
not. Those who prize the higher occupations of intel-
lect see with regret the tendency of our present social
system, both in England and America, with regard
to opinion upon the end and use of knowledge, and the
purpose of education. Of the thousands who, in each
year, take their station in the different parts of busy life,
by far the greater number have never known real mental
exertion ; and, in spite of the variety of subjects which
are crowding upon each other in the daily business of
our elementary schools, a low standard of utility is gain-

ing ground with the increase of the quantity of instruc-

tion, which deteriorates its quality. All information be-
gins to be tested by its professional value; and the know-
ledge which is to open the mind of fourteen years old is
decided upon by its fitness to manure the money-tree.
Such being the case, it is well when any subject can
be found which, while it bears at once upon questions
of business, admits, at the same time, the application of
strict reasoning ; and by its close relation to knowledge
of a more wide and liberal character, invites the student
to pursue from curiosity a path not very remote from
that which he entered from duty or necessity. Such a
subject is the theory of life annuities, which, while it
will attract many from its commercial utility, can hardly
fail to be the gate through which some will find their

XVi PREFACE,

a person whose ambition it is to walk in the brightest
boots to the cheapest insurance office, he has my pity:
for, grant that he is ever able to settle where to send his
servant, and it remains as difficult a question to what
quarter he shall turn his own steps. ‘The matter would
be of no great consequence if persons desiring to insure
could be told at once to throw aside every prospectus
which contains a puff: unfortunately this cannot be done,
as there are offices which may be in many circumstances
the most eligible, and which adopt this method of ad-
vertising their claims. If these pompous announce-
ments be intended to profess that every subscriber shall
receive more than he pays, their falsehood is as obvious
as their meaning; if not, their meaning is altogether
concealed.

Public ignorance of the principles of insurance is the
thing to which these advertisements appeal: when it
shall come to be clearly understood that in every office
some must pay more than they receive, in order that
others may receive more than they pay, such attempts
to persuade the public of a certainty of universal profit
will entirely cease. To forward this result, I have en-

deavoured, as much as possible, to free the chapters of |

this work which relate to insurance offices from mathe-
matical details, and to make them accessible to all edu-
cated persons. Whether they act by producing convic-
tion, or opposition, a step is equally gained: nothing
but indifference can prevent the public from becoming
well acquainted with all that is essential for it to know on
a subject, of which, though some of the details may be
complicated, the first principles are singularly plain.

August 3. 1838.

eg eS en

ADEE SCG Sees Be a

CONTENTS.

CHAPTER IL

On the Notion of Probability and its Measurement; on the Province
of Mathematics with regard to it, and Reply to Objections - Page 1

CHAPTER II.
On Direct Probabilities - - . ax = - = 30

CHAPTER III.
On Inverse Probabilities - S ~ fe e

Cr
09

CHAPTER IV.

Use of the Tables at the end of this Work - » ~ - 69
CHAPTER V.

On the Risks of Loss or Gain - oer o s - ¥8
CHAPTER VL

On common Notions with regard to Probability - . - IR

CHAPTER VIL
On Errors of Observation, and Risks of Mistake - - - 18

CHAPTER VIIL

| On the Application of Probabilities to Life Contingencies - - 158
CHAPTER IX.

: On Annuities and other Money Contingencies - ~ - 181
CHAPTER X,

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On the Value of Reversions and Insurances « = ie

XVill OONTENTS.

CHAPTER XI.
On the Nature of the Contract of Insurance, and on the Risks of
Insurance Offices in general . Page 237

CHAPTER XIi.
On the Adjustment vf tne Interests of the different Members in
an Insurance Office - . - e - 267

CHAPTER XIII.
Miscellaneous Subjects connected with Insurance, &c. a - 294

APPENDIX.

APPENDIX THE FIRST.

On the ultimate Chances of Gain or Loss at Play, with a particular
Application to the Game of Rouge et Noir - . ~ eas |

APPENDIX THE SECOND.

On the Rule for determining the Value of successive Lives, and of
Copyhold Estates - .. - - - Xv

APPENDIX THE THIRD.
On the Rule for determining the Probabilities of Survivorshir - xxii

APPENDIX THE FOURTH.
On the average Result of a Number of Observations - XXiV

APPENDIX THE FIFTH.

On the Method of calculating uniformly decreasing or increasing
Annuities - - - - - - XXvVi

APPENDIX THE SIXTH.
On a Question connected with the Valuation of the Assets of an In-
surance Office - . a “ - - XXX1

Table I. ” ° m » S<Xxtv
Table II. . ie . ~ XXXvVili .

ON

PROBABILITIES.

CHAPTER I.

ON THE NOTION OF PROBABILITY AND ITS MEASURE-
MENT 5 ON THE PROVINCE OF MATHEMATICS WITH
REGARD TO IT, AND REPLY TO OBJECTIONS.

~“Wuen the speculators of a former day were busily
employed in constructing celestial tables for the use of
prophets, or investigating the qualities of bodies for
the manufacture of gold, no one could guess that they
were accelerating the formation of sciences which should
themselves be among the most essential foundations of
navigation and commerce, and, through them, of civilis-
ation and government, peace and security, arts and liter-.
ature. That good plants of such a species require the
warmth of mysticism and superstition in their early
growth is not a rule of absolute generality, for there are
eases in which cupidity and vacancy of mind will do
as well. Cards and dice were the early aliment of
the branch of knowledge before us; but its utility is
now generally recognised in all the more delicate branches
of experimental science, in which it is consulted as the
guide of our erroneous senses, and the corrector of our
fallacious impressions. And more than this, it is the
source from whence we draw the means of equalising the
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Bis ESSAY ON PRVBABILITIES.
aiccidlents of life, and contains the principles on which
jt is found practicable to induce many to join together,
and consent that all shall bear the average lot in life of
the whole. But the ill educated offspring of a vicious
parent is frequently fated to bear the stigma of his de-
scent, long after his own conduct has created the good
opinion of those who know him. The science which I
endeavour, and I believe almost for the first time, to ren-
der practically accessible in its higher and more useful
parts to readers whose knowledge of mathematics ex-
tends no farther than common arithmetic, is still often
considered as foreign to the pursuits, and dangerous in the
conduct, of life. It is said to be necessary only to gam-
blers, and calculated to excite a passion for their worthless
and degrading pursuit. This refers to its practical and
moral consequences : with regard to its title to confidence,
it is often supposed to rest upon pure conventions of an
uncertain order, and to depend for the connection of
results with principles upon the higher branches of ma-
thematics ; things understood by very few, and frequently
distrusted, if not by those who have reached them, by
those who have passed some way up the avenue which
leads to them. All these impressions must necessarily
be removed before the theory of probabilities can occupy
its proper place ; and it is, therefore, my preliminary
task to meet the arguments which arise out of them:
There is an indefinite dislike in many minds to all know-
ledge which they cannot reach ; it may tend to remove
this if I show that results, at least, are very easily at-
tained, and methods practised: but the notion that
asserted knowledge is not knowledge must be met by
preliminary reasoning, and imperfect as it must neces-
sarily be, considered as a view of the subject, it may
yet afford the means of dwelling on the first principles
to a greater extent than is usually done in formal treatises
on recognised subjects.

Human knowledge is, for the most part, obtained
under the condition that results shall be, at least, of that
degree of uncertainty which arises from the possibility of

SS ore

}

PE eR aA

INTRODUCTORY EXPLANATIONS, 3

their being false. However improbable it may be, for in-
stance, that the barbarians did not overturn the Roman
empire, we do not recognise the same sort of sensible cer-
tainty in our moral certainty of the fact which we have in
our knowledge that fire burns, or that two straight lines
do not enclose space. And we perceive a difference in the
quality of our knowledge, when any alteration takes place
in our circumstances with respect to exterior objects.
That fire does burn is more certain than the account of
the fall of Rome: that fire yet to be lighted wild burn
may or may not be more certain than the historical fact,
according to the temperament and knowledge of the in-
dividual. And thus we begin to recognise differences
even between our (so called) certainties ; and the com-
parative phrases of more and less certain are admissible
and intelligible. It is usual to begin the subject by
saying that our certainties are only very high degrees of
probability. This is not practically true at the outset ;

yet so far as deductions can be made numerically.

with respect to our impressions of assent or dissent, it
will be shown to be correct so to consider the subject.

We have a process to go through before we can arrive

at such a conclusion, as follows: — When a child is
born, there is a certain degree of force which we allow
to the assertion that he will die aged 50. To it we
answer that it may be, but that that particular age is
unlikely compared with all the rest, though, at first
sight, as likely as any other. If the assertion be made
of two children, that one or other will die aged 50,
we readily admit that our “it may be, but it is not
likely,”’ is no longer the same assertion as it was before.
it is of the same sort, but not of the same strength: the
assertion is more probable, and wherever we have the
notion of more and ess, we feel the possibility of an answer
to the question, ‘‘ how much more or less? ” and which
we should produce if we knew how. First impressions
would induce us to suppose it twice as probable that the
assertion may be made of one or other of two children,
as of one alone; and so on. Let this false measure (for
B 2

4, ESSAY ON PROBABILITIES.

such it is) remain ; we are not here considering what is
the proper measure, but whether we can conceive the
possibility of a measure or not. Let the preceding me-
thod of measurement be admitted ; and let us ask how
we stand with regard to the same assertion, predicated
of one or other of a million of children born together.
The answer is, we feel quite certain, that many of them
will die at the age of 50, Supposing humanity to en-
dure 50 years, we feel as confident of the truth of the
assertion, as we do that Rome was taken by Alaric, or
that fire will burn. Without entering into the very
different sources through which conviction comes to us,
we put four propositions together : —

The Romanem- | Two straight | Fire will | Of 1,000,000 of

pire was over- | lines cannot burn. | children born,some
turned bynorth- | enclose a will die aged fifty,
ern barbarians. space. if the race of man

last fifty years.

and, we ask, if you were to receive a certain advantage
upon naming a truth from among these four assertions,
what would guide your choice? There is certainly a
little difference in the impressions of assent with which
we regard the four; but whether. it be of any real
strength, we may test in this way: — Supposing the
benefit in question to be 1000/., would you not let
another person choose for you, almost at his pleasure,
and certainly for a shilling ?

On this we remark, firstly, that by it we feel sensible
of our assent and dissent to propositions derived in very
different ways, being a sort of impression which is of
the same kind in all. To make this clearer, observe
the following: — A merchant has freighted a ship, which
he expects (is not certain) will arrive at her port. Now
suppose a lottery, in which it is quite certain that every
ticket is marked with a letter, and that all the letters
enter in equal numbers. If I ask him, which is most
probable, that his ship will come into port, or that he
will draw no letter if he draw, he will apswer, unques-
tionably, the first, for the second will certainly not hap-

es ste i

INTRODUCTORY EXPLANATIONS. §&

pen. If I ask, again, which is most probable, that his
ship will arrive, or that he will, if he draw, draw either
@ Oe, OF Cy 08. or 2, or y, or 2, he will answer, the
second, for it is quite certain. Now suppose I write the
following series of assertions : —

He will draw no letter (a drawing supposed).
He will draw a.

He will draw either a or 0.

He will draw either a, or b, or c.

ie will draw either a or b OF seccsseee OF ry.
He will draw either a or 5 or .,...006. OF Y OF %

and making him observe that there are, of their kind;
propositions of all degrees of probability, from that which
cannot be, to that which must be, I ask him to put the
assertion that his ship will arrive, in its proper place
among them. ‘This he will perhaps not be able to do,
not because he feels that there is no proper place, but
because he does not know how to estimate the force of his
impressions in ordinary cases. If the voyage were from
London Bridge to Gravesend, he would (no steamers
being supposed) place it between the last and last but
one: if it were a trial of the north-west passage, he
would place it much nearer the beginning ; but he would
find difficulty in assigning, within a place or two, where
it should be. All this time he is attempting to compare
the magnitude of two very different kinds (as to the
sources whence they come) of assent or dissent ; and he
shows by the attempt that he believes them to be of the
same sort. He would never try to place the weight of
his ship in its proper position in a table of times of high
water.

We also see, secondly, that the impression called cer-

tainty is of the character of a very high degree of

probability. Out of 1,000,000 of children born, it is

certain some will die aged 50. But by gradual pro-

gression, our unassisted judgment makes us _ believe

that we may correctly say that it is 1,000,000 times as
B 3

an

6 ESSAY ON PROBABILITIES.

probable the assertion will be true of one or other out of
1,000,000 as of one alone. The method of measuring
is wrong, but that is here immaterial; suffice it
that, come how it may, the multiplication of the degree
of assent implied in “ there is a remote chance of it”
is found to give that which is conveyed in “ we are
quite sure of it.” We have thus a sort of freezing and
boiling point of our scale of assent and dissent, namely,
absolute certainty against on the one hand, absolute
certainty for on the other hand, with every description
of intermediate state.

Thirdly, we have proposed two ascending scales of
assertions, in both of which first impressions would
make us suppose the probability of the second is double
that of the first, that of the third treble, and so on, as
follows : —

A child born will die aged fifty. | a must be drawn.
Of two children born, one or | a or 6 must be drawn.
other will die aged fifty.
Of three children born, one or | a, or 6, or c must be drawn.
other will die aged fifty.
&e. &e. &c. &e. &e. &C.

Now it will hereafter be positively proved that our
notion is correct in the second case, but incorrect in the
first; or at least that it cannot be correct in both.
Even then, if we should fail in assigning positive mea-
surements, we may succeed in drawing useful distinctions.
When we imagine two things to have a point of re-
semblance which they have not, it is worth while to in-
vestigate methods of correction, even though we cannot
assign how much the two properties differ which we
supposed were alike.

The quantities which we propose to compare are the
forces of the different impressions produced by different
circumstances. The phraseology of mechanics is here
extended: by force, we merely mean cause of action,
considered with reference to its magnitude, so that it is
more or less according as it produces greater or smaller
effect. It is one of the most essential points of the

INTRODUCTORY EXPLANATIONS. |

subject to draw the distinction we now explain. Pro-
bability is the feeling of the mind, not the inherent
property of a set of circumstances. It is frequently
referred to external objects, as if it accompanied them
independently of ourselves, in the same manner as we
imagine colour, form, &c. to abide by them. ‘Thus we
hold it just to say, that a white ball may be shut up in
a box, and whether we allow light to shine on it or not,
it is still a white ball. And if we were to translate the
common notion, we should also say that in a lottery of
balls shut up in a box, each ball has its probability of
being drawn inseparably connected with it, just as much
as form, size, or colour. But this is evidently not the
case: two spectators, who stand by the drawer, may be
very differently affected with the notion of likelihood in
respect to any ball being drawn. Say that the question
is, whether a red or a green ball shall be drawn, and
suppose that A feels certain that all the balls are red,
B, that all are green, while C knows nothing whatever
about the matter. We have here, then, in reference to
the drawing of a red ball, absolute certainty for or
against, with absolute indifference, in three different
persons, coming under different previous impressions.
And thus we see that the real probabilities may be dif-
ferent to different persons. The abomination called
intolerance, in most cases in which it is accompanied by
sincerity, arises from inability to see this distinction.
A believes one opinion, B another, C has no opinion at
all. One of them, say A, proceeds either to burn B or
C, or to hang them, or imprison them, or incapacitate
them from public employments, or, at the least, to libel
them in the newspapers, according to what the feelings
of the age will allow ; and the pretext is, that B and C
are morally inexcusable for not believing what is true.
Now substituting* for what is true that which A be-
lieves to be true, he either cannot or will not see that it

“* The refusal of this substitution is what soldiers call the key of A’s position:
he himself sees the absurdity of his own arguments the moment it is made;
and he is therefore obliged to contend for a sort of absolute truth external
to himself, which B or C, he declares, might attain if they pleased.

B 4

8 ESSAY ON PROBABILITIES.

depends upon the constitution of the minds of B and C
what shall be the result of discussion upon them. Let
it be granted that the intellectual constitution of A, B,
and C is precisely the same at a given moment, and
there is ground for declaring that any difference of
opinion upon the same arguments must be one of moral
character. Granting, then, that it were quite certain A is
right, he might be justified in using methods with B
and C which are reformative of moral character ; that
is to say, granting that state punishments are reform-
ative of immoral habits, as well as repressive of im-
moral acts, he would be justified in direct persecution.
But to any one who is able to see with the eyes of his
body that the same weight will stretch different strings
differently, and with those of his mind that the same
arguments will affect different minds differently — by
difference not of moral but of intellectual construction —
will also see that the only legitimate process of alteration
is that of the latter character, not of the former ;
namely, argument * and discussion. In the mean time,
we bring it forward as not the least of the advantages
of this study, that it has a tendency constantly to keep
before the mind considerations necessarily corrective of
one of the most fearful taints of our intellect.

Let us now consider what is the measure of proba-
bility. Any one thing is said to measure another when
the former grows with the growth of the latter, and
diminishes with its diminution. For instance, in the
tube of a thermometer, the height of the mercury above
freezing point (a line) measures the content of a cy-
linder ; not that a line is a solid, but twice as much
length beiongs to twice as much content, and so on.
Again, the content of the cylinder measures the quantity
of expansion in a given quantity of mercury (and in
this case not only measures, but is). Thirdly, the

* It is frequently asserted, that opinions dangerous to the existence of
public order must not be promulgated. This is a question distinct from the
one in the text, so far as it is political. If we grant no morals except expe.

diency, (which, it appears to us, is necessary for the affirmation of the pre.
ceding,) the answer is, simply, that persecution is ineffective. .

INTRODUCTORY EXPLANATIONS. 9

quantity of expansion measures the quantity of heat

which produces it.

The exactness of mathematical reasoning depends
upon that of our knowledge of the circumstances em-
ployed. No theorem about triangles, for instance, is
true of any approach to a triangle such as we make on
paper ; but only more and more nearly true, the more
nearly we make our lines lengths without breadths, and
straight. Similarly, we cannot apply any theory of pro-
babilities to the circumstances of life, with any greater
degree of exactness than the data will allow. But as in
geometry we invent exactness by supposing the utmost
limits of our conceptions attainable in practice, so in the
present case we begin by reasoning on circumstances de-
fined by ourselves, and require adherence to certain
axioms, as they are called, meaning propositions of the
highest order of evidence.

Axiom 1. Let it be granted that the impression of
probability is one which admits perceptibly of the
gradations of more and less, according to the circum-
stances under which an event is.to happen.

Axiom 2. Let it be granted that when one out of a
certain number of events must happen, and these
events are entirely independent of one another, the
probability of one or other of a certain number of
events happening must be made up of the probabi-
lities of the several events happening. For instance,
in the lottery of letters, in which there are 26 inde-
pendent possible events, the probability of drawing
either a, b, c, or d is made up of the probabilities
of drawing a, of drawing 6, of drawing c, and of
drawing d, put together.

The latter axiom may excite some discussion ; but we
must observe that it is the uniform practice of mankind
to act upon it, which is a sufficient justification ; for
what are we doing but endeavouring to represent that
which actually exists? With regard to the value of each
chance, suppose that one of the letters is a prize of 26/.,
and that the 26 letters have been bought, If I buy

10 ESSAY ON PROBABILITIES.

up all the vested interests at less than 1/. a piece, I am
certain to gain; if at more, I am certain to lose. 1/,
a piece is what I ought to give for each, if I buy all:
it is the universal practice to consider that 1/. a piece is
still the value, if I buy a part. To say this is in fact
to say that the force of the impression called certainty
should, in this case, be considered as made up of 26
equal parts, each of which is to be considered as the
representative of the impression of probability which a
right-minded man would derive from the possession of
one ticket.

On this I have to remark, 1. That so soon as any
notion receives the exactness of mathematical language,
though it be thereby not altered, objections are taken to
it. The reason is, that we frequently not only use ex-
pressions which can be rendered quite exact, but also
fairly act upon them as if they were exact, but not be-
cause we consider them exact. Why does the lottery
ticket of the preceding instance bear the character of
being exactly worth 1/.? Not as any consequence of
the accuracy of the preceding process, supposing it ac-
curate, but because we do not know why we should
exceed rather than fall short of it. It appears to me
that many of our conclusions are derived from this
principle, which is called in mathematics the want of
sufficient reason. <A ball is equally struck in two dif-
ferent directions, the table being uniform throughout.
In what direction will it move? In the direction which
is exactly between those of the blows. Why? No posi-
tive reason is assignable (experiment being excluded) ;
but from the complete similarity of all circumstances on
one side and the other of the bisecting direction, it is
impossible to frame an argument for the ball going more
towards the direction of one blow, which cannot imme-
diately be made equally forcible in favour of the other.
The conclusion remains, then, balanced between an in-
finity of possible arguments, of which we can only see
that each has its counterpoise. Now whether we adopt
the above conclusion as to probability for its exactness.

eee

INTRODUCTORY EXPLANATIONS. iI

or for its want of demonstrable inexactness one way
more than the other, it is still a principle of human
action, and as such is adopted. Many writers on pro-
bability speak of it as being a maxim which, if it were
not adopted, ought to be. Certainly, such an assertion
has some strong arguments in its favour ; but with me
they would not outweigh the importance I should attach
to exact deduction from the conceptions which actually °
prevail.

Let the prospect of drawing any given letter be
of a degree of force represented by 1, all the several
prospects being equal. Then 2 is the chance of draw-
ing one or other out of any given pair; and so on
up to 26, which is here the representative of certainty.
But if the lottery had 50 letters, the prospect of draw-
ing a given letter would no longer be represented by
1; or if so, the certainty of drawing one out of 50
in the second would be represented by 50, while the
certainty of drawing one out of 26 in the first is repre-
sented by 26. Now certainty, absolute certainty, should
have the same representation whatever contingencies it
may be supposed to be compounded of. If a man be
sure of 100/., it matters nothing whether his certainty
arise from the announcement of a prize in a lottery of
1000 tickets, or of a legacy to which 20 other people
were looking forward. ‘To use a common phrase, a man
can but be certain ; and therefore it would be desirable
to use the same symbol for certainty in all cases. Let
this symbol be unity or 1; then in the first lottery the
chance of any given letter is represented by =',, and in
the second by =!;. Similarly the chance of 1 out of 10
given letters in the first lottery is 4°, and in the se-
cond, 4°.

Now I pause upon this result, which, in fact, con-
tains all the theory I shall be obliged to use; grant
this, and you can be constrained, by demonstration, to
admit all the rest as simple logical consequences. A
writer on this subject, therefore, must take care not to
let an opponent of its principles choose his own ground of

12 ESSAY ON PROBABILITIES.

attack, so as to wait until he can take advantage of the
length of a deduction, or of the mathematical character
of the steps. Do you admit, 1. That a certainty, if you
have it, of drawing a 10/. prize in a lottery, is precisely
the same thing whether there be 100 or 1000 tickets ?
and 2. That if there be 3 white balls and 17 black in a
lottery, of which either white ball is to be a prize, you
’ are compelled to regard your chances of success and
failure with impressions of which it is reasonable to
suppose the force to be as 3 to 17; or to say, “ the
degree in which I fear failure is, to my degree of hope
of success, in the proportion of 17 to 3.” If you say
this, it matters nothing whether you say it because you
feel the correctness of the proposition, or because you
feel a want of data to deny it in one way more than in
the opposite. _ Provided only that you do not deny it,
your occupation of opponent is gone; for all that suc-
ceeds is merely a mathematical use of this mathematical
definition. In the words of the ritual, Speak now, or
ever after hold your tongue.

But it may be asked, with regard to the mathematical
part of this subject, What is the province of the science
of calculation? Are we, because we reject the higher
mathematics, entirely without evidence; or can we ob-
tain any thing like conviction of the truth of our me-
thods? Now it happens unluckily for objectors, that
the duty of mathematics in this science is very much
more simple in character than the same in astronomy,
mechanics, optics, music, or any other part of mathe-
matical physics. For in the whole of these sciences,
we have principles, as well as results, deduced by long
trains of mathematical reasoning ; whereas, in the
science before us, we ask nothing of mathematics but the
abbreviation of long numerical operations. For in-
stance : — “ If bodies move round another body, circu-
larly, and so that each body, in its own circle, describes
equal lengths in equal times, and if, moreover, the
squares of the times of revolution are in the same
proportion as the cubes of the distances, then it follows
that the cause of motion can be nothing but an attractive

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Le ef ee a ee A i SES ain EMO Oe ee eS te en ee es

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INTRODUCTORY EXPLANATIONS, 13

force directed towards the central body, which, for dif-
ferent distances, changes inversely as the square of the
distances.” This is well knoWn to be a fundamental
part of the system of astronomy which has enabled one
century to do more towards correct prediction of the
state of the heavens than the twenty centuries which
preceded it; and yet the apparatus of mathematics
which is required to establish this result, which is of the
nature of a principle, is enormous. But in the present
subject we shall establish all our principles without the
aid of any more mathematics than is contained in arith-
metic ; and when we draw upon the science, it shall be
for nothing but abbreviation of long processes. The
principle upon which mathematical abbreviation fre-
quently proceeds is this: that where the calculation of
a few results materially aids the production of a great
many more, it is advisable to calculate a multitude of
results, to arrange them in convenient tables of reference,
and to publish them ; so that by means of one person
taking a little more trouble than would otherwise fall to
his share, all others may be saved labour altogether.
Mathematical tables are frequently nothing but the re-
sult of labour performed once for all ; but it also some-
times happens, that the principle on which the labour
is performed can be exemplified by a familiar case of it.
We shall take that of logarithms as an instance.

Every table of logarithms is an extensive table of com-
pound interest. Not to embarrass ourselves with frac-
tions, let us take a table of cent. per cent. compound
interest. We have then the forlawsne: amounts of 1/. in
1, 2, 8, &c. years :—

Yrs. Am. | Yrs..| Am. i Yrs. Am. Yrs. Am.
O 1 7 128 # 14 16564 § 21 2097152
1 g 8 256 415 82768 } 22 4194304
2 4 9 512 | 16 65536 | 23 8388608
3 8 } 10 | 1024 4 17 | 181072 § 24 | 16777216
4/16 ;11 | 2048 | 18 | 262144 } 25 | 33554432
5 | 562 $12 | 4096 § 19 | 524288 | 26 | 67108864
| 6 | 64 £18! 8192 § 20 |1048576 | 27 1134217728

14 ESSAY ON PROBABILITIES.

The property of this table is, that if we wish to
multiply together any two numbers called amounts, we
have only to add togetMer the number of years they
belong to, and look opposite the sum in the table of
years. Thus, 11 and 12, added together, give 23;
2048 and 4096, multiplied together, give 8,388,608.
The reason is as follows: if 1/.in 11 years yield 20481,
and if this 2048/. be put out for 12 years more, then,
since 1/. in 12 years yields 4096/., 2048 times as much
will yield 2048 x 4096/.; or the amount in 11+12
years is the product of the amounts in 11 and 12 years.
The only reason why the preceding table is not in the
common sense of the words a table of logarithms, is, that
its construction leaves out most of the numbers. We
can deal with 2048 and 4096, but there is nothing
between them. The remedy is, to construct a table of
compound interest, at such an excessively small interest,
that a year shall never add so much as a pound through-
out. Certain considerations, by which the table may be
_ shortened, but with which we have here nothing to do,
make it convenient to suppose such a rate of interest, that
1/. shall increase to 10/. in not less than 100,000 years,
at compound interest. Or we may suppose interest pay-
able 100,000 times a year, and say, let the whole yearly
interest be 1000 per cent. per annum. ‘Taking the first
supposition, we have a part ot a table of logarithms as
follows :-—

Am.

1000
1001
1002
1003

Vrs.

Am,

Yrs.

Am

| Yrs.

Am.

Yrs.

300043
300087
300130
300173

5232
5233
5234
5235

371867
371875
371883

9997
9998

371892

9999
10000

399987
399991
399996
400000

10001
10002
10003
10004

400004
400008
400015
400017

&e. | &c. | &e.| &c.

This is the light in which a common reader may view
a table of logarithms. Let 1 increase to 10 at compound
interest in 100,000 equal moments, then 1 will become
5234 in 371,883 such moments; and soon. We can
thus manage to put down every number, within certain

INTROPRUCTORY EXPLANATIONS, £5

limits, as an amount; and thus, within those limits, we
reduce all questions of multiplication and division to
addition and subtraction, by reference to the tables.

We thus perceive a simple principle applied with
much labour, but such as is performed once for all.
The notion above elucidated was the first on which
logarithms were constructed ; in time came more easy
methods. We now take another abbreviation which is
perpetually occurring in our subject. It is the multi-
plication of all the successive numbers from 1 up to
some high number ; thatis, the continuation of the pro-
cess following. Let [10], for instance, represent the
product of ali the numbers, from 1 up to 10, both in- |
clusive, or let

[10] stand forl x2x3x4x5x6x7x8x9x 10=3628800

[l]is 1; [2] is2; [8] is 6; [4] is 245 [5] is 1203 [6]
is 720; [7] is 5040; [8] is 40,320; and soon. This
labour becomes absolutely unbearable when the numbers
become larger; thus, [$0] contains 33 places of figures,
and [1000] contains 2568 figures. But, nevertheless,
we cannot deal with problems in which there are 1000
possible cases without knowing, either nearly or exactly, —
the value of [1000]. It will, however, be sufficient
to know this value very nearly ; within, say, a thou-
sandth part of the whole ; that is, as nearly as when, the
answer of a problem being 1000, we find something be-
tween 999 and 1001. We now put before the reader
who can use logarithms a rule for this approximation,
with an example; intending thereby to show the reader
who does not comprehend the process how mathematics
enter this subject in the abbreviation of tedious comput-
ations,

Ruute.— To find very nearly the value of [a given
number], from the logarithm of that number, subtract
*4342945, and multiply the difference by the given
number, for a first step. Again, to the logarithm of the
given number add *7981799, and take half the sum, for
a second step. Add together the results of the first and

16 ESSAY ON PROBABILITIES.

second steps, and the sum is nearly the logarithm of the
product of all numbers up to the given number inclusive.
For still greater exactness, add to the final result its ali-
quot part, whose divisor is 12 times the given number.
Exampie.— What is [30] or 1X2 X3X...... x 29

x 30?

log. 30 1°4771213 1°4771213
*4342945 °7981799

Subtract 1:°0428268 2)2-2753012 Add.
30 wae

en 1°1376506 Second step.

Multiply 31-284804 First step.
1°137651 Result of second step.

Ce ee

32°422455 log. of result.

The result has, therefore, 33 places of figures, of
which the first six are (nearly) 264,518 ; or, if this be
increased by its 300th (12 times 30) part, or about
735, the result is 265,253, followed by 27 ciphers ; or

the approximate result is —

265253000000000000000000000000000

The true result is —
965252859812191058636308480000000

and the error is not so much of the whole, as one part
out of 500,000.

In this way, we are able to do with more than
sufficient nearness, and in a few minutes, what it
would take days to arrive at by the common method,
and with much greater risk of error.

If we wish to find the product of all the numbers,
say from 31 to 100, both inclusive, we find [100] and
[30] approximately, and divide the first by the second.
We shall represent this by [31,100]: thus,

[7,15] stands for 7x8x9x10x11x12x13x%14x15

But, though we can thus simply put the logarithmic
computer in possession of a great acquisition of power
we can get through much the greater part of our task
without such a process, by means of a table of which

INTRODUCTORY EXPLANATION. 17

it is not possible to explain either the principle, or the
reason of its utility, to any but a mathematician. We
can only explain its mere construction, as follows: —

B

a
Renae anne
} " a eo

A N \ wie.

Let A B be one (inch, for example) ; and take an inde-
finitely extended line A X, perpendicular to A B: from
A towards X let a curve be conceived to be described,
so that every ordinate NP shall be connected with its
abscissa AN, by the following law. Measure AN in
inches and parts of inches; and multiply the result by
itself; and the product by °4342945. Find the num-
ber to which this product is the common logarithm,
and divide 1°1284 by the result. The quotient is the
fraction of an inch in NP, and in the table We find,
not N P, but the area AN P B expressed as a fraction of
a square inch. The curve itself is what is called an
asymptote to A X, continually approaching, but never
reaching, AX: and the whole area, AX being continued
for ever, is one square inch. To this table I shall have
continual occasion to refer: into it, in fact, is condensed
almost the whole use I shall have to make of the higher
mathematics.

I have thus drawn the distinction between the prin-
ciples of the subject, as derived from very obvious
results of self-knowledge, and the principles of mathe-
matics, applied merely to the abbreviation of the tedious
operations which large numbers require. I now pro-
ceed to the several assertions which have been made
upon the nature and tendency of the subject.

I, That it is not true. The whole weight of this
assertion, and of all arguments in its favour, falls
entirely upon the method of measurement in page 11.,
and ultimately upon the second axiom, in page 9.
Again, as we are most unquestionably justified in say-
ing that it is more probable we shall draw one of the

c

18 ESSAY ON PROBABILITIES.

two, a or b, than that we shall draw a, the argument
must be directed against the method of measurement,
not against the possibility of a measure: for wherever
more or less are applicable terms, twice, thrice, &c.
must be also conceived to be possible, whether we can
ascertain how to find them or not. But no other
method of measurement has ever been proposed, nor, in
truth, have the assertors been aware that they could be
brought to such close quarters, but have generally ob-
jected to the theory as a whole, without any particular
knowledge of its parts. It will be time enough to
refute their notion, when they begin to be so particular
that refutation becomes possible.

II. That it is not practical. By this it is either
meant, (for practical is one of the words employed in
shifting an argument, which are sometimes so con-
venient), that it has not been reduced to practical
form, og else that it is not capable of being so reduced ;
or perhaps that it is not useful. The working results
hitherto obtained may be divided into: —1. The method
of obtaining probabilities. —2. The method of estimating
the probability of more or less departure from the
results indicated by the main branch of the theory as
most probable. The first has been frequently made
practical ; the second not hitherto, except to mathema-
ticians. That the whole can be made practical, I hope
to establish by the contents of this work. To the asser-
tion that it is not useful, we oppose: —1. The unanimous
opinion of astronomers, (meaning thereby persons capa-
ble of applying the subject to astronomy ) that the
exactness of our present knowledge is very much owing
to the application of it, and their uniformly continuing
to use it in the deduction of results from the necessary
discordances of observations. —2. The extent to which
it has been applied in the very choicest view of
the word practical, (which frequently means money-
making) in concerns which now employ many millions
sterling. — 3. The light in which it is regarded by a
very large majority of those who have studied it, as a

INTRODUCTORY EXPLANATION. 19

corrector of false impressions, and indicator of just and
necessary, though not always perceptible, distinctions. —
4. The beauty of the study itself, considered merely as
a speculation, and as a method of exercising certain
powefs of mind, which might otherwise lie useless. —
5. The necessity of informing the public as to the real
nature of the occupation called gambling, and of the
class of men who live by it; the latter being persons
who are using knowledge of these principles success-
fully, to the daily loss and ruin of those who are not
aware of what constitutes unequal play. If such argu-
ments be not sufiicient to counterbalance a simple asser-
tion, to the extent of making it worth while to decide
the question by an examination of the subject itself, we
may safely dispute the utility of any branch of know-
ledge.

III. That it has a tendency to promote gambling.
Those who make this objection generally use the common
signification of the term gambling ; and the motives for
this pursuit are, in their view, either the pleasure of
suspense, acting as a stimulus to a mind weary of its
own vacancy, or the desire of gain. On the first notion,
the assertion is self-destructive; it amounts to saying
that knowledge which diminishes suspense, by giving a
better view of the circumstances, has a tendency to
promote gambling, by affording the pleasure arising
from suspense. So far as the theory of probabilities
bears upon gaming in general, its tendency is to convert
games of chance into something more resembling games
of skill. Now games of skill are seldom made the ve-
hicles of very high play. So far, then, the tendency of
our study is to substitute the satisfaction of mental
exercise for the pernicious enjoyment of an immoral
stimulus. With regard to the desire of gain, we may
safely admit that those who are already actuated by this
motive in an undue degree, will sometimes be led to
-gamble by knowing how to do so properly ; and just in
the same manner some of them will be led to make
forgery the means of increasing their store, from knowing

c 2

20 ESSAY ON PROBABILITIES.

how to write. But the fear that those who seek
a livelihood by what is commonly called gambling,
which always means cards, dice, or horse-racing, &c.,
would be much increased in number, if at all, by such
a pursuit as the mathematical appreciation of proba-
bilities, seems to me grounded upon a want of know-
ledge of human nature. Putting out of view the
tendency of all serious thought to lead the mind toa
perception of its own resources, and to furnish methods
of employing time; and not even considering that the
demand for this baneful excitement is controlled by the
opinion of society, and lessened by the amount of edu-
cation: there still remain the means of showing that
the balance is in favour of a study of the theory of
probabilities, even as a preventive of this very gambling
which it is said to provoke. Nemo repenté fuit tur-
pissimus: and, it will be one of our objects to show,
that the person who lives by gaming, deserves the
strongest form of the adjective. No one ever said to
himself, I have not played hitherto, but I will begin
henceforward to make it my trade. A young man who
is ruined by play in the first instance, or who, at least.
has begun by courting as an amusement what he ends
by requiring as an occupation, is the subject of which
a gambler is made. Now, suppose that all those who
have been ruined by play had been trained to under-
stand the true nature of their pursuit. Let it be
granted that some of them are so fond of acquisition,
that it is only necessary to point out a plausible method
to insure their following it: yet we must grant, on the
other side, that there will be some who can be per-
suaded that when they play against a bank or a
gamester, they are almost certain of playing on very
unequal terms, which is never what they contemplate
and intend. The only question is, which of the two
numbers will be the greatest; 1. Of those who be-
come gamesters prepense, or, 2. Of those who either
take a total or a partial warning; the latter in a
degree sufficient to insure a fair chance for them-

pee
ie

INTRODUCTORY EXPLANATION. 21

selves, The thoughtlessness of youth will be urged
against my opinion, that the latter number would be very
much the greatest. I reply, that, comparatively speak-
ing, and with respect to maturer years, young men
are thoughtless ; but, absolutely speaking, they are not
so with respect to dangers of which they know the
risks, The ill success of others does not deter them, be-
cause they attribute it to fortune ; and, because they have
superstitions hanging about them with respect to luck
which are tolerably prevalent in all classes. They think
they are trying their luck, as the phrase is; but if
they could be convinced that it is not their luck
which they are trying, but only a fraction of it,
their opponent having the rest in his pocket, they
would show themselves in this, as in other matters,
averse to risks in which it is more than an even
chance against them. They come to the consideration
of the subject fraught with wrong notions, which have
been carefully instilled as preventives. The character
of a gambler is represented as dishonest, in the com-
mon sense of the word. That is to say, the term
gambler is confounded with that of sharper, meaning a
person who would mark a card, or load a die. They
find the falsehood of this notion in their commerce
with the world: gamblers show themselves in the face
of day who really appear to be, and are, men of ho-
nour in the common sense of the word, and who would
scorn any under-handed proceeding, under ordinary
temptation at least. What then becomes of the pre-
vious warning? It is proved to be false in an essential
part; and is therefore lost altogether. Add to this
that the principle of the occupation is misrepresented :
admonition is given against trying fortune, instead of
proof that fortune is not tried. A proposition is ad-
vanced which is an absurdity: equal play is supposed,
and yet it is maintained that the luck will generally be
against the inexperienced. Skill is considered as only
adding to the chances against the unskilful, instead of
creating a certainty, and arguments drawn from a single
. c 3

92 ESSAY ON PROBABILITIES.

game, which are really good, are applied to collections
of many games, with regard to which they are not ap-
plicable. I will leave it to any one to say, whether the
considerations pointed out in the succeeding pages have
the tendency to promote the pursuit of fair gaming as
a means of profit.

With regard to gambling as a stimulus, it must be
observed, that the passion has every where subsided with
the increase of education and occupation. If the his-
torians who write for schoolboys could spare a little space
among their interminable accounts of kings, treaties,
battles, to insert some account of the manners of the
several ages of Europe, it would be matter of surprise
that the universal rage for games of chance, should have
left any time for the (so called) great actions which fill
the books. The wars of the middle ages would be
looked upon as belonging only to one particular class of
the stimuli by which the universal vacancy was sought
to be filled up. From the old Germans, who played
away, to one another, their wives, their children, and
lastly themselves, down to the time * of the French re-
volution, the continent of Europe (and during a part
of the time, Great Britain, though in a less degree,) —
gives, comparatively to ourselves, the notion of succes-
sive races devoted to gambling throughout the upper
class, the only one upon whose occupations we get fre-
quent details.

IV. That the basis of it is an irreligious principle.
There is a word in ‘our language with which I shall
not confuse this subject, both on account of the dis-
honourable use which is frequently made of it, as an
imputation thrown by one sect upon another, and of the
variety of significations attached to it. I shall use the
word anti-deism to signify the opinion that there does
not exist a Creator, who made and sustains the universe.
The charge is, that a theory of probabilities (called
chances) is necessarily anti-deistical, because it refers

* Quand, avant la révolution francaise, les états de certaines provinces
- aient assemblés, on y jouait un jeu terrible, et tel que Vendroit ot il se

enait, dans la c- devant province de Brétagne, s’appellait Venfer. — Dict.
he Jews (Encyc. Meth.) 1792.

INTRODUCTORY EXPLANATION. 983

events to chance. Various modifications of this asser-
tion present themselves, but they may all be referred
either to that just made, or to a tendency argument of
the same character. All the sciences have had to
encounter this aspersion, each in its turn ; but it is to be
remarked, that philosophy and philosophers have always
been charged with the worst thing going. The believers
in sorcery never failed to attribute an intimate connection
with infernal spirits to all who investigated nature in
any form: the believers in anti-deism follow in their
steps. There is in the proposition above mentioned, a
shifting of the meaning of terms: it has been customary
to designate anti-deism as the opinion that the world was
made by chance, meaning, without any law or purpose
existing ; but the word chance*, in the acceptation of
probability, refers to events of which the law or purpose
is not visible. Thus a great part of the application of
this subject has been destroyed by successive discoveries,
When the observatory at Greenwich was founded, the
chance errors of observation were large in the fixed stars.
Nothing could be said but that there was a deviation
which appeared of one sort in one observation, and of
another in another, without visible law or order. Brad-
ley’s discoveries removed much of this, that is, pointed
out law where law was not seen to exist before. Im-
provement of instruments and methods of observation
has still more distinguished the error into parts with a
visible, and parts with an invisible, cause. As an
answer to the species of argument employed, nothing
more is necessary: those who can, may consider this
science as not bearing on religion, either in one way
or the other, so far as anything in the preceding argu-
ment is concerned, or in the explanation which is no
more than necessary for an answer. But there is a view
of the subject, and that one most indispensable, which

* Generally speaking, the abstract singular term chance has the anti-
de.stic meaning, while the plural chances is used for the several possibilities
of an event happening. Thus Hume says : — ‘‘ Though there be no such
thing as chance in the world..... there is certainly a probability which
arises from a superiority of chances.”

c 4

24, ESSAY ON PROBABILITIES.

better deserves to be made a fundamental principle, than
an incidental answer to a futile objection. The past
contains our grounds of expectation for the future:
Why? Because we cannot help supposing that there
were causes which produced the past, and which con-
tinue to act. If there be any one to whom this is not
a truth, he cannot proceed with us one step. Suppose
that 100 drawings out of an urn all give white balls, the
presumption is very strong that the 101st will give a
white ball also. But if there neither be, nor ever were,
some reason why the balls so drawn should be white
rather than black, that is, if the event be pure chance,
then the 100 drawings afford no presumption whatever —
that the 101st will be either white or black. So far
then as we have yet gone we have the following positive
and negative conclusion :

The theory of probabilities
absolutely requires, in its fun-

The theory of probabilities,
so far as considerations of ab-

damental principles, the rejec-
tion of the notion that pure
chance can produce any two
events alike; that is, it pre-
sumes causation and order of
some kind or other, that is,
providence of some kind or

solute necessity are concerned,
neither denies nor asserts, in
whole or in part, any thing
whatsoever respecting the mo-
ral or intellectual character of
the providence which it re-
quires to be granted.

other.

From the preceding we may be certain that no con-
clusion in any way leading to natural religion, however
faint, is tacitly assumed in the premises. If there-
fore such a conclusion should follow legitimately, it
stands upon a basis of absolute security. This is not
often the case in arguments drawn from nature in gene-
ral, on account of the mixture of considerations with
which the mind is affected by them. When we speak
of the vastness, the regularity, and the permanency of
the solar system in general, the very immensity of the
argument would prevent the mind from being aware whe-
ther there was or was not either an appeal to constitutional
feeling, distinct from reason, or even an assumption of
the question in the manner of deducing it. The cele-

INTRODUCTORY EXPLANATION. 25

brated work of Paley may be considered as a treatment
of the following syllogism : — “ If there be contrivance,
there was design; but there is contrivance, there-
fore there was design,” the minor of which is proved by
appeal to observation. But the author refers the
opponent to the beauty and ingenuity of the methods in
which the contrivance is brought about: to the general
effect on our notions of what is done, compared with what
we could do. It may very often be discovered what the
real tenor of an argument is, by observing what would
refute it: now imagine an individual possessed with the
notion that he could execute * better contrivances, and
Paley’s argument must (to him ) be imagined to be ineffec-
tive.t Itappears to me that the result of the treatise in
question is this: ‘‘ If there be a contriver, he must be
one of infinite power and intellect.” But the argument of
contrivance against chance cannot, from the complication
and non-numerical character of the instances, be illustrated.
by any reference to what might have been if chance had
prevailed. Taking, for example, the chamois, as the
result of a contrivance for the support of animal life on
frozen mountains, we have no method of comparing the
chamois of design with any notion that we can form,
and call the chamois of chance. But where a consider-
ation is pure number, we then have other ideas, of the
homogeneity of which with that in question, we feel
assured: and we can absolutely try the question with
chance in precisely the same manner as we try it in the
common affairs of life. Let us assume, as we must,
that a number produced by chance alone (in the anti-
deistical sense of the word,) might as well have been
any other as what it is. And further, let us require
before we grant intelligence and contrivance, not merely
the presence of an adaptation which would have been
unlikely from chance alone, but two such phenomena,

* Such as is said actually to have struck Alfonso of Portugal, when the
Ptolemaic theory of the heavens was explained to him.

+ The fault of most treatises on Natural Theology is to draw the reader’s
attention from the mere design, to the complication and ingenuity of the
design. The Bridgwater Treatises have a consistent title, and it is worthy
of remark, that this was the doing of the testator himself,

26 ESSAY ON PROBABILITIES.

perfectly distinct from each other considered as phe-
nomena, each of which might have existed without
the other, and both tending to the same object, which
would have been defeated by the absence of either.
Let it also be granted, to fix our ideas, that we admit as
proved, a proposition which has a hundred million to one
in its favour.

This being premised, and laying it down as our object
to show that the necessary results of the theory of pro-
babilities lead to the conclusion that the existence of
contrivance is made at least as certain, by means of it,
as any other result which can come from it, we proceed
to state a consequence :— The action of the planets upon
each other, and that of the sun upon all (the most cer-
tain law of the universe), would not produce a perma-
nent * system unless certain other conditions were fulfilled
which do not necessarily follow from the law of attrac-
tion. The latter might have existed without the former,
or the former without the latter, for any thing that we
know to the contrary.t Two of these conditions are,
that the orbital motions must all be in the same direc-
tion, and also, that the inclinations of the planes of these
orbits must not be considerable. Granting a planetary sys-.
tem which is what ours is, in every respect except either
of these two, and it is mathematically shown that such
a system must go to ruin: its planets could not preserve
their distances from the sun. Neither of these phe-
nomena can be shown to depend necessarily on the
other, or on any law which regulates the system in
general. For any thing we know to the contrary, then,
they are distinct and independent circumstances of the
organisation of the whole. Now let us see what are
the phenomena in question : —

* Permanent, not liable so to change as to destroy the organisation of
the parts. Ifthe earth could ever approach so near to the sun that all the
water should be vaporised, the permanency of the system would be de-
stroyed, so far as our planet is concerned.

¢ The only way in which we can guess any two things to be independent.
It must be remembered, as a result of the theory, that, ot things perfectly
unknown, the probability of their coming to act, when known, against an

argument, is counterbalanced by the equal probability of the future dis.
covery being on the other side.

INTRODUCTORY EXPLANATION. oF

I. All the eleven planets yet discovered move in one
direction round the sun.

I], Taking one of them (the earth) as a standard, the
sum of all the angles made by the planes of the orbits of
the remaining ten with the plane of the earth’s orbit, is
less than a right angle, whereas it might by possibility
have been ten right angles.

Now it will hereafter be shewn that causes are likely
or unlikely, just in the same proportion that it is likely
or unlikely that observed events should follow from them.
The most probable cause is that from which the observed
event could most easily have arisen. Taking it then as
certain that the preceding phenomena would have fol-
lowed from design, if such had existed, seeing that they
are absolutely necessary, ceteris manentibus, to the main-
tenance of a system which that design, if it exist, actu-
ally has organised, we proceed to inquire what prospect
there would have been of such a concurrence of circum-
stances, if a state of pure chance had been the only ante-
cedent. With regard to the sameness of the directions
of motion, either of which might have been from west
to east, or from east to west, the case is precisely similar
to the following :—- There is a lottery containing
black and white balls, from each drawing of which it is
as likely a black ball shall arise as a white one; what
is the chance of drawing eleven balls, all white. Answer,
2047 to 1 against it. With regard to the other question,
our position is this. — There is a lottery containing an
infinite number of counters marked with all possible dif-
ferent angles less than a right angle, in such manner that
any angle is as likely to be drawn as another; so that in
10 drawings the sum of the angles drawn may be any
thing under 10 right angles. What is the chance of 10
drawings giving collectively less than 1 right angle?
Answer, 10,000,000 to 1 against it. Now what is the
chance of both of these events coming together? An-
swer, more than 20,000,000,000 tu 1 against it. It is
consequently of the same degree of probability that there
has been something at work which is not chance, in the

28 ESSAY ON PROBABILITIES.

formation of the solar system. And the preceding does
not involve a line of argument addressed to our percep-
tions of beauty or utility, but one which is applied
every day, numerically or not, to the common business
of life.

Now whether what precedes amounts to the means
of producing rational conviction, it is not necessary
for me to stop and inquire. The question is, how
do the results of this theory affect those moral and
intellectual considerations which it has been stated to
have a tendency to overthrow? It matters nothing to
my present purpose how much of the preceding a reader
will admit ; for the point, considered with reference
to the objection before us, is this : -— Does the preceding
deduction weaken the probability of the existence of an
intelligent creative power? for if not, the objection is
overthrown, and whatever strength is conceded to belong
to the reply is so much addition to other arguments
in favour of the same conclusion. Let us suppose a
reader so much biassed against the higher parts of
mathematics that he does not feel any confidence in
the united work of all ages and countries amounting
to more than a millionth of certainty. There still re-
mains 20,000 to 1 in favour of the conclusion above
stated, after weakening the preceding by introducing a
probability that all the exact sciences may be wrong,
such as his state of mind requires. With respect to the
bearings of the theory, we may now add the follow-
ing to the statement in page 24.

Applying the first principles of the theory of proba-
bilities, by means of mathematics, to the phenomena of
the universe, it is a necessary conclusion that the ex-
istence of something which combines together different
and independent arrangements to produce an end which
could not, ceteris manentibus, be produced without
them, must be added to the notion of a Providence, in-
telligent or not, which is required in the first prin-
ciples.

With regard to the existence of a revelation from the

INTRODUCTORY EXPLANATION. 29

Supreme Being, this theory leaves the question exactly
where it found it; and the same of all questions of his-
torical evidence. If we were to assume fictitious data,
we night, as in all other sciences of inference, produce a
consequence which should be as true as the premises,
standing or falling with them. The science itself is
the deduction of the probability in a complicated case
from the probability in a known and simple case. But
where is the known and simple case in the historical
question? In valuing testimony, no theory of the
method in which conflicting evidence should be com-
bined will help us to the original value of the several
parts of it, any more than an investigation of the
method of solving an equation will help us to a know-
ledge of the particular equations which apply in any
given case. ,

The two great theoretical questions before us are :—
I. What is the measure of probability? II. What is
the way of using it? — The necessary preliminary to
application is, the result of the measurement in a case
to which the method of measuring can be applied, and
has been applied. The mistakes which have arisen from
confounding these considerations are numerous. For in-
stance, tell me how many times per cent. a given man will
be wrong in his judgment, and I can tell you exactly,
positively, and mathematically, how much more likely a
unanimous jury-(not starved) is to have arrived at a true
decision, than another in which the voices are 8 to 4.
But that does not put me one step nearer to ascertaining
what is the per centage of erroneous conclusions in
the judgments of a single individual. The miscon-
ceptions just alluded to are equally prevalent with regard
to all the sciences; a person who studies astronomy is
frequently asked what the moon is made of.

Much of the objection, religious or not, made against
probability in general, is connected with the notion
already mentioned, (page 7.) that it is a fundamental
quality of events, external to ourselves, which is under
consideration: on which the rational feeling must be,

30 ESSAY ON PROBABILITIES.

that there is no such thing. The term probability is
as difficult to explain as gravitation: and the method of
proceeding is the same with regard to the properties of
both. We cannot tell what they are, in simpler terms,
but we know them by, or rather trace and define them
by, their manifestations. In both, we first see a com-
pound result, depending upon the patient as well as the
agent. In the case of the mental phenomena, we can-
not decompose the effect produced, still less ascend a
step, and find any of the laws which regulate the hu-
man disposition to doubt or expect. I shall conclude by
again reminding the reader, that the impression produced
by circumstances upon his own mind is the thing in
question ; and that nothing can be more liable to cause
confusion than a lurking notion that the results of theory
are anything more, before the event arrives, than a re-
presentation of the relative force of his own impressions,
as they should be if unassisted reason could follow the le-
gitimate consequences of some simple and universally
admitted principles.

I proceed in the next chapter to develope the leading
rules of the science.

CHAP. if.

ON DIRECT PROBABILITIES.

We now proceed on the supposition that the probability
of an event is measured by the fraction which the
number of favourable cases is of all that can happen.
Thus, if there be 20 white balls and 27 black (20+ 27
or 47 in all), the probability of drawing a white ball is
measured by $7, and that of drawing a black ball by 214.
We shall say that these probabilities ave %2 and 24,
Nothing is more common than to substitute a measure

ON DIRECT PROBABILITIES. 31

for the thing itself: thus we talk of a temperature of
60°, when, in fact, each 1° only means a certain length
on the tube of the thermometer. It will illustrate this
to take a case in which we do not confound the thing
and its measure; in the barometer, for instance, we
never say that the air’s weight is 30 inches, but that the
height of the barometer is 30 inches.

The technical words, probability and improbability,
must now be considered as meaning the same thing in
different degrees. If there be only one white ball out
of a thousand, we usually say that to draw the white
ball is possible, but not probable. We now speak of it
as having asmall probability, namely —;;!;5 ; we might
say it has a great improbability, namely ~°°99., but this
phrase is not customary. The moment we obtain either
numerical measures, or distinctions which are not verbal,
the distinctions which are verbal frequently become su-
perfluous and inconvenient. Thus in the art of book-
keeping, profit and loss never appear as separate words,
but only as part of a complex term profit-and-loss,
meaning, one or the other, according to the side of the
account on which the item is found. ‘To a mercantile
reader, we should say that probability means probability-
and-improbability the first or second, according as its
measure is greater or less than}. When the number
of favourable and unfavourable cases is the ee say 50
of each, the probability of the event is °°; or 4; and in
this case we say in common life that there isa “balance
of probabilities, or that the event has an even chance.

By the word chance with the article (a chance) we
mean one single way in which an event may happen, as
when we say that every white ball adds a chance to the
prospect of drawing a white ball. In the first instance
above, the chances of white and black are as 20 to 27.
It is also usual to say that the odds are here 27 to 20 in
favour of black against white,

Questions on probability are twofold in character :
1. Where we know the previous circumstances and re-
quire the probability of an event. 2. Where we know

32 ESSAY ON PROBABILITIES.

the event which has happened, and require the proba-
bility which results therefrom to any particular set of
circumstances under which it might have happened.
The first I call direct, and the second inverse, questions.

We must begin with direct questions, though the in-
verse precede the direct in practice. For, as we know
the whole range of possible cases in hardly one instance,
we cannot proceed with points which have reference to
matters of life until we know what presumptions arise
with respect to the whole, from observation of a part.

In direct questions of probabilities, the event may
either be simple, that is, depending on one indivisible
event ; or compound, consisting of several events which
may happen together. Thus, suppose four events, either
of which may happen, and call them A,B,C, D.
Knowing the circumstances of each, I may ask the fol-
lowing questions, every one of which states an event
simple or compound.

1. What is the chance of A. 2, What is the chance
that one shall happen and only one. 3. What is the
chance that one or more will happen. 4. What is the
chance that one at least will happen, and one at least
will not. 5. What is.the chance that a given pair, and
no others, will arrive. 6. What is the chance that a
given pair at least will arrive. 7. What is the chance
that some pair or other will arrive, but only a pair.
8. What is the chance that a pair at least, will happen,
&c. &c. All these are most evidently distinct questions
when they are clearly proposed; but it is almost as
evident that they are very liable to be confounded.

The mathematical definitions and theorems which
will be necessary for our purpose, are the following : —

1. A permutation means a number of cases selected
out of all possible cases, in some particular order ; so
that different arrangements of the same things make dif-
ferent permutations. Thus if all the possible cases be
A,B, C, and D, we have the following

Permutations of one out of four, A, B, C, and D.

33

Permutations of two out of four, AB, BA, AC, CA, AD,
DA; BC; CB, BD, DB, CD, DC.

Permutations of three out of-four, ABC, ABD,BAC,BAD,
ACB, ACD, CAB, CAD, ADB, ADC, DAB, DAC,
BCA, BCD, CBA, CBD, -BDA, BDC, DBA, DBC,
CDA, CDB, DCA, DCB.

Permutations of four out of four (different arrangements of
four), ABCD, ABDC, BACD, BADC, ACBD, ACDB,
CABD, CADB, ADBC, ADCB, DABC, DACB, BCAD,
BCDA, CBAD, CBDA, BDAC, BDCA, DBAC, DBCA,
CDAB, CDBA, DCAB, DCBA.

To find the number of permutations of one number
(in) out of another (n), begin with the whole number
(n) and write down as many numbers, reckoning down-
wards. as there are units in the number (m) which are to
be in each permutation: then multiply all together.
For instance: How many permutations are there of 4
out of 12, The answer is.

be M22 AR IG 2 See

ON DIRECT PROBABILITIES.

11880

The following table will furnish examples or serve
for reference.

| 10 i 9 8 7 6 5 /4.|8 |2 [1 |
1} 10 | 9 8 7 6 15 |a|s left |
2) 90 | 72 56 | 42 | 30 | 20 |1ale {2

3) 720 504 336 | 210 | 120 60 joe | |
4| 5040 | s024 | 1680 | 840 | 860) 120 24

5| 30240 | 15120 | 6720 | 2520 | 720] 120
6| 151200 | 60480 | 20160] 5040 | 720
7| 604800 | 181440 | 40390 5040 |
s| 1814400] 362880 | 40320 |
9| 3628800] 362880 |

10] 3628800 | |

4 ESSAY ON PROBABILITIES.

Thus the number of permutations of 6 out of 9 is
opposite to 6 under 9, or 60480.

2. By a combination is meant the same thing as a
permutation, except only that arrangement is no part
of the idea. Thus, of all the permutations of three out
of four in A,B,C, D, the following, ABC, BAC,
ACB, CAB, BCA, andCBA,are all the same
combination. ‘Thus we have
a permutation : a selection in a certain order.

a combination : a selection without reference to order.

To find the number of combinations, divide the
number of permutations by the product of all the
numbers up to the number in each combination. Thus,
how many combinations can be made of 4 out of
12? The answer is found by

dividing 12 x1llxlOx9
by lx 2x 3x4

The process may always be shortened by striking
common factors from the dividend and divisor. Thus,
if we wish to know the number of different hands which
can be held at whist (combinations of 13 out of 52) we
must
divide 52. 51. 50. 49. 48. 47. 46. 45. 44. 43. 42. 41. 40.

by 1, 62-8. 4. °5. -6e 7. 8. 9. 10. 13.122

The shortest way is to decompose every number in
both into its factors, when it will appear that all the
factors of the divisor are found among those of the di-
vidend ; as follows :—
1B. 2.2. 17.3..5.562.467. 2 28.2.2 47. 2:23.: 5.9.8. 2438.

43. 2.7.3. 41. 2.2.2.5.
1, 2. 3. 2.2. 5. 3.2. 7, 2.2.2. 3.3. 5.2. 11. 2.2.3. 184.5

} which gives 495.

and the factors which must be multiplied to give the
result are

V7. 7. 47, 23. §& 43. 2.7. 41. 2.2.2.5.
or 17. 47. 23. 43. 41. 35. 560 giving 635013559600.

When the number in each combination is more than
half of the whole number, the rule may be shortened.

ON DIRECT PROBABILITIES. 35

Thus, if ! ask how many combinations of 21 can be taken
out of 25, I do in effect ask how many combinations
of 4 may be taken. For there are just as many ways
of taking 21 as there are of leaving 4.

The following table corresponds to the one preced-
ing: —

10 9 | 8

1 10 ie,

@1 45 36 | $051.0). 13K L1G kth ae diad

3 | 190 84 | 56.1 $5 | PO IO et

41210 {126 | 70 51 is) s+ 4

9 10 1

10 1

Thus the combinations of 5 out of 9 are 126 in
number.

3. When the same event may be repeated in a per-
mutation, the number of permutations is the product
of as many numbers, all equal to the whole number of
possible events, as there are of events in each permutation.
Thus, of four events, A, B,C, D, the number of per-
mutations of two, with repetition, is4 x 4 or 16 ; with-
out repetition 4 x 3 or 12. The additional 4 in the
first case, are AA, BB, CC, and DD. The number
of permutations of three, with repetition, is4 x 4 x 4,
or 64; without repetition, 4 x 3 x 2 or 24, Of the addi-
tional 40, 10 are where A onlyis repeated ; namely, AA B,
AAC,AAD, ABA, ACA,ADA,BAA,CAA,

D2

36 ESSAY ON PROBABILITIES.

DAA, that is nine where A is repeated twice, and
AA A, one, where A is repeated three times. Ten
more are those in which B is repeated twice, &c.

4. The number of combinations in which there are
repetitions must be determined without rule, in every
particular case. Suppose I wish to know how many com-
binations, including repetitions, can be made of four
out of seven. Let the seven be A, B, C, D, E, F, G.

I. The number, without repetition, is 35.

II. Those in which A is repeated twice, and no other,
are evidently as many as the number of pairs in B, C,
D, E, F, and G, that is, 15. Thus we have AA BC,
AACD, &c. There are as many in which B is re-
peated twice, &c., so that 7 x 15 or 105, is the num-
ber in which one only is repeated twice.

III. The number in which A is repeated three times
is evidently 6, and the number in which one or other is
repeated three times only, is 6 x 7 or 42,

IV. The number in which one is repeated four
times, is 7.

V. The number in which two are repeated twice, is
as many as the number of pairs in 7, or 21.

Consequently, the whole number is made up of 35
105, 42, '7, and 21, or it is 210.

We have not so much to do with combinations allow-
ing repetition, as with permutations of the same kind.

To avoid the perpetual occurrence of long phrases ©
for simple ideas, I shall use the following abbrevia-
tions: — By P f 4,20 } is meant the number of per-
mutations of 4 out of 20, without repetition: by PP
§ 4,20} the same with repetition. By C {4,20}
is meant the number of combinations of 4 out of 20,
without repetition: by CC {4,20} the same with
repetition. Again, by [4,20], as in page 15., is meant
the product of all numbers from 4 to 20, both inclu-
sive; and, by 7!°, asin algebra, is meant ten sevens
multiplied together. Thus, a reference to the preceding
rules will show the following : — —

ON DIRECT PROBABILITIES. 37
P {5,12} ig [12,8], pP{ 5,12} is 19
cf 4 25} is (25, 22] divided by [1, 4]

5. Tf there be an event composed of several others in
succession, of which the first may happen in, say 10
different ways, the second in 14, and the third (let there
be but three) in 6, then the compound event must be
one out of 10 x 14 x 6, which is the number of dif-
ferent ways in which it may happen. When all the
component events may happen i in the same number of
ways, this reduces itself, both in principle and rule, to
the case of permutation with repetition.

I now proceed to some problems requiring nothing
but a knowledge of the measure of probability and the
preceding rules. Any event of known circumstances
may be familiarly represented by a lottery of balls of
different colours. Thus, suppose ten Russian ships,
twelve French, and fourteen English, are expected in
port, or may have arrived. Let one be as likely to ar-
rive as another, and suppose it known that two have
arrived, but not of which nation they are. Let a cer-
tain advantage accrue to A, if they should happen to be
Russian and French, of which he is desirous of selling
his chance immediately. The question is precisely the
_ following : — Let there be a lottery of green, white, and
red balls, 10, 12, and 14 in number. Let two have
been drawn, and suppose a certain advantage to arise to
A, if they be green and white. What should be given
to A, certain, in consideration of his chance P

Here we must first consider the number of ways in
which two can be drawn. All the balls are 36 in num.
ber, and the whole number of pairs is 36 x 35 + 2
or 630. Of these a green ball (1 out of 10) may be
paired with a white ball (1 out of 12) in 120 different
ways, ssp gg the probability of his gaining the
advantage is 749 or 54. The probability against it is
therefore | +1; or out of every 21 possible events, 4 are
in favour of, and 17 against, the advantage. It is there-
D $

38 ESSAY ON PROBABILITIES.

17 to 4, or 44 to 1, against the advantage. If the con-
tingent gain were 21/. the value of the chance would be
4]., as will hereafter be more fully shown.

Of all the pairs which can be drawn, there are
10 x 12=120 green and white. 10x 9+2=45 both green.
12 x 14=168 white and red. 12 x 11+2=66 both white.
14x 10=140 red and green. 14x 13+2=91 both red.
The sum of these is 630, as it should be. The reader
should explain to himself the reason of the difference
of process. The least probable ,case is both green,
the most probable white and red Nothing is more
common than the idea that the event most likely to
happen, which is compounded of two or more events, is
a repetition of the event which is individually the most
probable. This is true of repetitions: red being most
probable, it is more probable that red should be repeat-
ed than that white should be repeated, or that green
should be repeated ; but that two white ones should be
drawn is not so probable as that a white and red should
be drawn. The drawing, whatever it ‘may be, is con-
sidered independently of succession — and this makes an
important difference. If the balls were drawn succes-
sively, both red is more probable than white followed
by red, or than red followed by white, but not more
probable than the chance of one or other, which is s the
preceding case.

I here also take occasion to notice the common
error, that because an event is more probable than any
other, it is the one to be looked for. The question
ought to be, Is that event more probable than some one
or other out of all the other events which may happen P
If ten persons engage in a competition, with an equal
chance of success, and if two of them, A and B, enter
into partnership, it is now more probable that the firm
of A and B will win than that C will win, or that D
will win, &c. But the chances against the firm are
still 8 to2 or 4 to 1. If a hundred halfpence be
tossed up into the air, the result which is more probable
than any other, is 50 heads and 50 tails. But com-
mon sense will tell us that the chances of this result are

ON DIRECT PROBABILITIES. 3

©

very small, owt of the whole, and calculation would con-
firm it.

Before laying down any more specific rules, I shall
take an instance of a somewhat complicated deduction,
in order to show that we do not really require any
other principle than is contained in the measure of
probability. Suppose A and B to play at whist against
Cand D. A begins, holding ace, king, and queen of
trumps, and these trumps only: what right has he to ex-
pect that, by playing them out in succession, he will force
all the trumps of the other party? That is to say, ten
trumps being distributed among B, C, and D, and any
single card having the same chance of belonging to either
of the three, what is the probability that neither C nor
D holds more than three trumps ?

Firstly, as to the number of tenures, which are per-
fectly distinct. To find the number of ways in which any
number of distinct objects can be divided among any
number of persons, use the following RuLE: —

Multiply together numbers equal to the number of

persons as often as there are things to be divided among
them. Thus, to find in how many different ways ten
distinct cards can be divided among three persons,
find

3x3x3, &c. (ten threes) or 3!° which is 59049.

The question now is, how many of these 59049
ways favour the supposition that neither C nor D holds
more than three out of ten. Both together they cannot
hold more than six: if, then, we pick out any given
siz of the ten trumps, that set may be divided
among C and D in 2° or 64 ways. But of these, there
are two ways in which 0 and 6 may be held, twelve
ways for 1 and 5, and thirty ways for 2 and 4, none
of which must be included. It must be observed, that
in the last sentence, the six distinct ways into which
6 may be divided into parcels of 1 and 5 must be
doubled, because * each gives two of our cases, ac-

* It is important to observe that no duplication must take place on this
account, if the two numbers be the same; for instance, in dividing 6 into
pareels of 3,

Dn 4

40 ESSAY ON PROBABILITIES.

cording as C holds 1 and D holds 5, or C holds 5 and
D holds 1: and so of the other cases. Consequently,
out of a given set of six, there are 64 all but 44, that
is, 20 ways, in which, when they happen, A could force
all the adversary’s trumps. But every set of six yields
20 such cases: hence the ways in which C and D can
together hold six trumps, are 20 times C f 6,10} or
20 X 210, or 4200. Similarly, a given set of 5 trumps
may be held by C and D in 2° or 32 ways, of which 2
and 10 must be excluded. Hence 20 x C f 5,10
or 20 x 252, or 5040 sets are the number favourable
to the event in this case. Four trumps can be held
in 2* or 16 ways, of which 2 must be excluded, or 14
xC j 4,10 } that is, 2940, is the number of favour-
able cases. On the supposition that C and D together
held only 3, or 2, or 1 trump, no exclusions are neces-
sary, and the number of cases are 2° x C f 3,10 ¢
or 960, and 22 x C $2,10$ or 180, and 2 x
C 1,10 or 20; and there is one case in which C
and D hold no trumps. All the favourable cases are,
therefore, in number

4200 + 5040 + 2940 + 960 + 180+ 20+1 or 13341.

The chance of A being able to force all the adversary’s
trumps is +3344, or nearly 3! to 1 against it.

Given a fraction less than unity, and which has high
numbers in its terms, required a set of fractions which
shall be very nearly equal to it, and each of which shall
be nearer than any other fraction of the same order
of simplicity. Required, also, a near estimate of the
error committed in each case.

Ruz. First perform the process for finding the
greatest common measure of the numerator and deno-
minator.

ON DIRECT PROBABILITIRS. 4
1 1834i)59049(4—4
53364

2 5685)13341(2x44+1=9
11870
Qx2+1=5 1971)5685(2x 9-+4+4=22
3942
5x14+2=7 1743)1971(1x22+9=31
1743
7Xx7+5=54 228) 1743(7 x31 +22=239
1596
54x 1+4+7=61 147)228(1 x 239-431 =270
147
81)147(1
81
66 &e.

Opposite to the first quotient write 1, and proceed te
form columns out of the several quotients, in the fol-
lowing manner ; —

1 =1st Numerator. Ist Quotient = Ist Denomina-

tor.

9nd Qu.=2d Num. 2d Qu. x Ist Qu. + 1 = 2d

Den.

2d Num. x 3d Qu. + Ist Num. | 3d Qu. x 2d Den. + 1st Den.
=3d Num. =3d Den.

3d Num. x 4th Qu. + 2d Num*| 4th Qu. x 3d Den. + 2d Den.
= 4th Num. =4th Den.

4th Num, x 5thQu. + 3d Num. | 5th Qu. x 4th Den. + 3d Den.
= 5th Num. = 5th Den.

Any numerator multiplied by | Any denominator multiplied
the next quotient, and pro-| by the next quotient, and
duct increased by preceding product increased by pre-
numerator, gives succeeding ceding denominator, gives
numerator. succeeding denominator.

Thus, +, 2, 35, do vss, 3%, &e. &c., isa set of
fractions which approach nearer and nearer to 4+33-}.
The first is always too great, the second too small, the
third too great, the fourth too small: every odd one too
great, every even one too small.

Test of correctness. Take any two successive numer-
ators and denominators, multiplied crosswise ; they give
products which differ by unity.

42 ESSAY ON PROBABILITIES.

54 61 54 x 270=14580
239 270 239 x 61=14579
1

Estimation of the error. The error of 3} is less than
sis (4 x 9 = 36); that of § is less than +}5 (9 x
22 = 198); that of ,9; is less than ;4,5 (22 x 31 =
682), &c.

Th ‘pomed ctbsieeblactd is less than :
rror of ; ; s les
oy ae its Denominator That Den. x the next.

I now take the following problem: A die is thrown
time after time; in how many times have we an
even chance of throwing an ace. The common error
attached to this problem is, that since there are six
faces, it is most likely all will have come up in six
throws.

In the first throw there are six events, five of which
are unfavourable. In the first two throws, considered
as giving one event, there are 6 x 6 or 36 possible
cases, for every possible case of the first throw may com-
bine with any case of the second. But of these 36
throws, any one of the five unfavourables of the first
throw may combine with any one of the second throw,
and there are therefore 5 x 5, or 25 unfavourable com-
pound events. Hence the following table : —

One throw gives 6 cases ; 5 unfavourable

Two throws give 36 cases ; 25 unfavourable

RIOD dnsvenseseae BLG secacs 3 125 ..rccrcccsenieee KUNG
against. )

Bc ocaswounnce A906 scscss $1 OBS. coun. 0 ccekiunin (odds
turned in favour. )

oo Ee eee TUT@ icthie 3 3125 &ce.

Answer: There is not quite an even chance of doing it
in three throws, but more than an even chance of doing
it in four.

Ruxte. When the odds against success in one trial
are n to 1, then -'; of n. (or the nearest whole num-
ber to it) is about the number of trials in which there
is an even chance of one success: more correctly 19° of
n. ‘Thus, if it be 144 to 1 against a single attempt,

t
ON DIRECT PROBABILITIES. 43

there is about an even chance of one success in 100
throws.

A table of the number of trials (cdds being n. to 1)
in which there are various odds of succeeding once : —

69 | 13 to 1 | 264 | 3O0tol | 343 § 1000 to1 | 691
110 | 14to1 | 271 § 40 tol | 371 | 2000 to1 | 760
139 | 15 to1 | 277} 50to1 | 393} 3000 tol | 8v1
161 | 16 to 1 | 283} 60 tol | 411f 4000 to 1 | 829
17 to1 | 289 | 70to1 | 426{ 5000 tol | 852
195 | 18 to 1 | 294 | 80to1 | 439 | 6000 to 1 | 876
208 | 19to 1 | 300 | 90to1| 451 | 7000 tol | 885
220 | 20 to 1 | 304 {100 to 1 | 462 | 8000 to 1 | 899
230 | 21 to 1 | 309 [110 tol | 471 | 9000 to 1 | 911

22 to1 | 314 120 to 1 | 48) | 10,000 to 1 | 921

248 | 23 to 1 | 318 | 130 to 1 | 488
12 to 1 | 256 | 24tol | 322 $140 tol | 495

The method of using this table is as follows: —- Sup-
pose the odds to be 20 to 1 against success in a single
trial: required in how many throws it is 100 to 1 there
shall be one success (or more). Look in the table
opposite to “100 to 1,” and we see 462 ; multiply 20
by 462 and divide by 100 (which is always to be done)
this gives 924, or 92 is about the number of throws
required.

I now proceed to the method of compounding the
probabilities of single events, so as to find those of com-
pound events: that is, the way of methodising the re-
sults of actual inspection.

Let there be two events, one of which may happen in 7
ways, and may not happen in 5; and the other of which
may happen in 4 ways, and may not happen in 9:
whence the probabilities of the two events (A and B)
are =4, and =4;. Now, the compound event may hap-
pen in 12 x 13 different ways, for any case of the first
(12 in all) may come up with any case of the second
(13 in all), By similar reason, the compound case in
which both A and B happen may come up in 7 x 4
different ways ; hence the probability of A and B both
happening, is

1x4 7 4
omen cone Cee
jax ig Whichis the product of 3 and 13

(© CONT CrP 69 DD
Ch Oh Oh et ct ct ct er ct
coooseosgse
Pht ek ek ed ek et

—

~I

io)

ood
©
aot
oo
—oy

=

Rute. When events are perfectly independent, so

AA ESSAY ON PROBABILITIES.

that the happening of one has nothing to do with that
of the other, the probability that both will happen is
the product of the probabilities that each will happen.

This rule applies to any number of independent
events.

It is indifferent whether the events are to happen to-
gether, or one after the other: thus the chances of a
compound drawing out of two lotteries, one drawing
out of each, are the same whether two persons draw
simultaneously, or one person first draws out of one, and
then out of the other.

Examp.e. Let there be three lotteries, as follows : —

6 white v neat { 8 white
5 black 2 black 10 black

What is the chance of drawing from the three, white,
black, and white? The probabilities of ee events are,
°;, 3, and -8,, the product of which is 54°, or about 17
to 1 against the compound event.

The probability that A will happen, and B will not,
is the product of the chance for A, and that against B.
The probability that one will happen, and one only, is
the sum of the probabilities —1. that A will happen and
B will not; 2. That A will not happen and B will
happen. The probability that one or both will hap-
pen is the remainder when the probability that neither
will happen is subtracted from unity. The following
are evident results of the measure of probability : —

1, When either P, @, or R must happen, the sum of
their probabilities must be unity, which is always the
representative of certainty. Thus, if a lottery contain 7
white, 5 black, and 3 red, either bene black, or red
must be drawn, and =4, -3., and -3,, together make 1,
and the same if the events be more or less than three in
number.

2. When of two events, each excludes the other,
the probability that one or other of them will happen is
the sum of their probabilities. For to say that each
excludes the other, is to say that they are connected
eveuts, or different possible cases of the same set. Thus,

‘

ON DIRECT PROBABILITIES. AS

in the preceding lottery, if I draw white I cannot draw
black, and vice versd, there being one drawing only
supposed. Hence the probability of drawing black or
white is = += or +2. But suppose there had been
two lotteries, as follows :—

(7 white, 8 red) (5 black, 10 red),

and I draw in one without knowing in which I am
drawing. The individual probabilities of Pagan in the
two, if the lottery be known, are ;‘, and -°- as before ;
but the question, what is the chance of padre white
or black, and not red, is a different one from the pre-
ceding. If I draw without knowing in which lottery 1
am drawing, my position is the same as if one lottery
had been thrown into the other, and I had drawn from
both in one, giving
(7 white, 5 black, 18 red).

This union, which will readily be admitted to make no
alteration in the probabilities, is not admissible unless
the total number of balls in both be the same. For, in .
the mixture, the 8 red balls of the one and the 10 of
the other, are considered as severally of equal proba~
bility. But this they are not, unless the number of
possible cases in the two were the same. Suppose, for
instance, I had the following lotteries : —

(10 white, 10 red) (4 black, 5 red) ;

the probability of each ball of the first is =1,, but of
each of the second 4. Before I can draw a white ball,

two events must happen. 1. I must happen to select
the first lottery. 2.1 must happen to draw a isi
ball rather than a red. The cist of these are 4 and
4° or +5 consequently, 4 x + or 4 is my chance of a
white ball, But if I mix the two lotteries, that chance
will be 48, which is more than 1. The reason is, that I
have mixed together balls which had unequal chances of
being drawn, and treated them as if they were to have
equal chances. But it is evident, from the measure of
probability, that I do not alter the chances for an event

46 ESSAY ON PROBABILITIFS.

or the general chances which any set of circumstances
affords, if I multiply the chances for an event any num-
ber of times, provided I multiply the chances against it
as often. Reduce the preceding lotteries te common num-
vers of balls, by putting 20 balls for one into the second,
and 9 for one into the first. We then have —

(90 white, 90 red) (80 black, 100 red).
Now mix the two, and we have —
(90 white, 80 black, 190 red) ;

and the chance of a white is ,%.°,, or 4, that is, the pro-
blem is not altered by the mixture. And the same may
be shown in any other case.

Let us now suppose that the chance of A is 2 and
that of B 4. The chances against A and B are there-
fore } and 2.

| The possible cases are The probability of which is
That A and B shall both happen 2x# or }f
‘That A shall happen, and not B 2x# Or sf

_ That B shall happen, and not A 4x? or 3;
That neither shall happen. tx#2 or 3

One of these cases must happen, and the sum of the
chances is 31 or 1, each compound event being ex-
clusive of the others. Again, we find

{ Both or neither is 12
One or other, but only one §
One or other, or both if

a One or neither
The probability A or both }4 A or neither $;

B or both 45 B or neither
| A or neither, or both }$
UB or neither, or both f
The latter set of events does not consist of those which
are mutually exclusive. Any thing which happens falls
under six of them, that is, six of them must happen.
Thus, A, and not B, actually arriving, would secure any
gain which depended upon either of the following : —
One or other, but only one A, or both

One or other, or both A, or neither
One or neither A, or neither, or both,

ON DIRECT PROBABILITIES. 47

If we add popademah all the probabilities of the preceding,
we find 4%, or 6. This is the indication that every
possible event enters in six different ways into the
contingencies whose probabilities united amount to
six.

Exampie. Twelve halfpence, A, A,...Aj,o, are
thrown up, required the probability of all the cases
which can happen, and which we shall symbolise thus:
(H3T.) means that there are three heads and nine
tails. The chance of H or T in any one piece is 1;
consequently, the chance of the several pieces A, A,
&c., yielding each any Pee letter is ixt xi
X wn (twelve factors), or ;545- Now, the ‘4096 pos-
sible cases are thus classified : —

cole

H, T,,. and H,, T, happen in 1 case each.
1 ia 1] 11 ceesecceece 12 cases each.

H, a4 eeoeos is oe as eeseeceseeose cfe, 12 or 65 eeeeeoeeeo

H, Ty eeece ° H, T e8Bcervescss CG 35 12 or BQO Riese ade

H, Ts eeeese H, I, seeereveeses G 4, 12 or AOS ccasnares

*H, Ty,’ «0... o Hy Vg cdssecesesee ©} 55 12°F Or 799 ee
H, T, happens in 924 cases.

It appears, then, that the most likely individual re-
sult is He T,, against which, however, it is about 34
to 1. But if we ask for the probability either of this
or of a single variation on one side or the other, that is,
of the following event —

Either H; T,, or H, T,, or H, T; —

we find 792 x 2+ 924 or 2508"(more than half of 4096)
cases in which the event arrives. That is, the chances
are in favour of one or other of these arriving. As the
number of events increases, a given degree of nearness
to the most probable event becomes more and more
likely. To illustrate this: suppose 24 halfpence thrown
up, the notation remaining as before. The total number of
cases is now 22+ or 16777216: calculating the number

4S ESSAY ON PROBABILITIES.

of combinations of 1,2,3, &c. out of 24, we have the
following summary :—

H, T.4 and H,, T, happen in 1 case each.
Big Digg oonens wy Ag ‘inenabipee ves 24 cases each,
(Faas ere $e bg sevvesesoess ZIG scccoosreses
Phe! hes sbsces je Pline orp creer 2024. .cccsoseces
H, Ty « es we rere ry 10626 J, .ccccceos
Bg “Tyg sees: Fig Tis cscseccecses 4DSO® ce cctve die’
eee wee I! ear ne 134596 .....s050 se
Bly Ty ovene Hyg Ty soeerees wad: 7PACIO‘ wecsioccones
Bog hag coer Ehig Tg scponpenenee HORT L onspavaieoes
ee cig: EN eats 1807504 ..eceeee ae
cS ag Ree 5 las UB gi ag fo 1961256 isch ececeus
Bt ee Pe 2496144 35.

ia ss happens i in 2704156 cases.

The odds are now about 6 to 1 against the even di-
vision of the pieces into heads and tails. But let us
consider the same degree of departure from the most
probable case as we took before. An alteration in one
piece out of twelve answers to that of 2 pieces out of
24. Now the number of cases in which either Hy
T,,,0rH,,T,;,,orH,.T,.,01H,,T,,,orH,,T,,
arrives, and the odds of one or other of them, is com-
puted as follows :— i
H,, T,,0r Hy, Tyo 3922512 16777216 whole No. of cases
H,, T,3 0r H,, T,;, 4992288 11618956 No. favourable

Hoot. s1g61s6 4
——-___‘ 5158260 No. unfavourable
11618956
or it is now more than 2 to 1 in favour of the heads
lying between 10 and 14, both inclusive.

I now put together the principles on which we have
hitherto gone, adding two more, the first of which
(Principle III.) is obviously a consequence of the pre-
ceding, and the second of which will be presently ex-
plained.

Principle I. When all the ways in which an event
may happen are equally probable, the chance of its
happening is the number of ways in which it may
happen, divided by all the number of ways in which it
raay happen and fail.

~~

ON DIRECT PROBABILITIES. 49

Principle II. The probability of any number of in-
dependent events all happening together, is the product
of their several probabilities.

Principle IJI. he probability of two events arriving
together being known ; and also, that of one of them: the
probability of the other is found by dividing the first
mentioned probability by the second.

Principle IV. When an event may happen in several
ways, whether equally probable or not, the probability
of the event is the sum of the probabilities of its hap-
pening in the several different ways.

The best way of illustrating the last principle is by
beginning with one of the numerous errors into which
we may fall, either in proceeding towards it, or applying
it. Let there be an event which may happen in two
different ways, for each of which there is an even chance.
Then, according to the principle, it is certain (4 + }=1)
that the event must happen. In this there is nothing
inconsistent. Let there be a lottery containing ten white
balls, and let them be sub-divided into two sets of five
by amark. Then there are two ways of drawing a
white ball, for each of which there is an even chance ;
namely, I may choose a ball of one mark, or of the other.
But it is evidently certain that in this case a white ball
must be drawn. Now, suppose an event can happen in
three different ways, for each of which there is an even
chance. Then the event is more than certain (4 + 4
+ 4= 14); which is absurd. But the absurdity is in
the supposition : an event can only have an even chance
of happening in one particular way, when that way
involves half of the total number of individual cases of
the event ; and it is impossible that three different ways
of arriving can each contain half of the whole number
of possible cases. Consequently, when we have made a
calculation of the probabilities which different ways of
arriving give to an event, there has certainly been an
error if the sum of the probabilities exceed unity.

But when we throw three half-pence into the air, are
there not three different ways of throwing head, for each

E

50 ESSAY ON PROBABILITIES.

of which there is an even chance? If by H we here
mean a single H, the three events by which we propose
to-attain it are not H, H, H, but H TT, THT, and
TTH, the probability of each of which is 4 and, by the
application of the principle, } + 4+ 4, or 4, is the
probability of the event,—one single head. If by:
throwing a head we mean one head or more, the ways
under which it may be brought about are— one head’
only (involving the cases H T T, THT, T T H)—two
heads only (involving HHT, H TH, and TH H)—
and three heads (involving + HH.) The probabilities
of these are 3, 3, and 4, or 4 is the chance required.
Another error to which we are liable, is the wrong
estimation of the probability of the different cases. For
instance, a person is to go on until he throws H, and is to
win if he do it in less than five throws. There are then
five possible cases ; namely, H, TH, T TH, T T T Hi,
T TTT. In four of these cases he wins ; in the fifth he
loses. His chance of winning then appears to be +.
But this supposes the five cases to be equally likely,
whien is i true. tase several probabilities are +, 4,
t, zz and =1,: not 1, +, &c., as supposed. Consequently
the sum of ee probabilities which the several winning
cases actually have is 12, or it is 15 to 1 that he wins.
If we put together all the cases, which four throws pre-~
sent, thus,—
HHHH 5. HTHH 9 THHH ts. TTA
. HHHT 6: HLHT . 10. THHT 14. TTHT

HHTH 7. HTTH 11 THEE 15. TTTH
HHTT 8 HTTT 12. THTT ig. TITTs

all those which begin with H are 8 in number, those
which begin with T H are 4, with T T H, 2, and with
TTTH, one. Hence 8+ 4 +2 +41, or 15, is the
number of winning cases. But the argument against
us, is this: most of the preceding cases are impossible,
for the condition is, that the play shall stop as soon as
H occurs: so that, in fact, the only possible cases are,
Het, FTTH DT TH,-.T T TD, sees te to;
we must then represent the several events as follows :—

Aer

ON DIRECT PROBABILITIES, 51

Ist event ; sigan 5 sh hen and gives either H or T.
Probability of H Be

2nd event; does not ue happen, but is con-
tingent upon ‘the first throw being T ; and gives either
H or T. Probability + that the throw is made, + that
if it be made it gives H; probability ao ee throw is
made, and that, being made, it gives H, 4 x = 14,

5rd event; contingent upon the two first throws
giving T and T, of which ihe chance is 1. Probability
of winning at this throw, + x 4=1.

4th event ; cietaeck: ‘upon the three first throws
giving T, T, T. Probability of winning at this throw,
gxXo= is

It is obvious enough, when stated, that every cone
tingency must enter into the consideration of a question ;
so that if some of the circumstances depend upon the
manner in which preceding contingencies arrive, this
circumstance itself influences the method of proceeding.
If we wish to avoid the necessity of considering a con-
tingency the trial of which is itself contingent, that is, if
we wish to make a contingency certain, we must in-
troduce all the new events which such change of con-
tingency into certainty brings with it. The whole
problem is exactly the same as if we made the four
throws certain, and made the gain dependent upon one
head or more being thrown: but we revert again to the
former state of the question, if we agree to mark the first
H as the winner. In this point of view, the distinction
between the two is evidently immaterial.

Exampie. There are seven lotteries, as follows (W
means white, B black) :—

I Wy:1 ae

II. (3 W, 2P) Lia Ww rf 2
W, 5B) VI.

IIE. (1 W, 3 B) Law 1 B) VIL;
and the conditions of drawing are the following. I. is
drawn, and then II. or III., according as I. gives W or B.
If II. be drawn, then IV. or V. is to be drawn, according
as II. gives W or B. But if III. be drawn, then VI. or
E 2

“

I. (2 W,3 ®)|

52 ESSAY ON PROBABILITIES.

VII. is to be drawn, according as III. gives W or B.
What are the probabilities of the several possible
drawings P

If from I. we draw W (of which the chance is 2),
we proceed to II. If we still draw W (chance, 3), we
proceed to IV. And here the chance of W is 1.
Hence,

the chance of WWW is 2 x?xi=3

Computing all the other chances in the same way, we

get the following :—

1WWW?2 2 4=2 5. BWW? ] t=
2 WWBA 8 j= 8 6 BWB 2? } =}
3. WBW 2 2 2=<28 7% BBW 2 %4=%
4.WBB 2? 2 4=4 8 BBB 3 % = ih

The sum of all these is equal to unity, as it should
be, since one or other of these cases must happen. And
by reducing all to the common denominator 5. 5. 4. 6,
or 600, we have the following chances : —

1. WWW 22 | s. WBW £4, | 5. BWW 4, | 7. BBW 3218
2. WWB 2 | 4.WBB {| 6.BWB %| 8. BBB $4,

That all shall be white, 7 to 1 against (nearly),

Two white and one black, $ to 1 against (very nearly)

Two black and one white, an even chance (nearly)

All black, 10 to 1 against (nearly).

From what has been said in this chapter, no great
difficulty will be found in ordinary questions. The cir-
cumstances are supposed to be fully known, and the
probabilities will be found, of the strength which it
follows they must have, to those who admit the axioms
on which the measure of probability is founded,

ON INVERSE PROBABILITIFS. 53

CHAPTER III.

ON INVERSE PROBABILITIES,

In the preceding chapter, we have calculated the
chances of an event, knowing the circumstances under
which it is to happen or fail. We are now to place
ourselves in an inverted position: we know the event,
and ask what is the probability which results from the
event in favour of any set of circumstances under which
the same might have happened. This problem is fre-
quently enunciated as follows: — An event has hap-
pened, such as might have arisen from different causes :
what is the probability that any one specified cause did
produce the event, to the exclusion of the other causes ?
By a cause, is to be understood simply a state of things
antecedent to the happening of an event, without the
introduction of any notion of agency, physical or
moral. ‘

In order that we may secure a problem of sufficient
simplicity, we must limit the number of possible ante-
cedent states. Let us suppose that there is an urn, of
which we know that it contains balls, three in number,
and either white or black, all cases being equally pro-
bable: that is, before any drawing takes place, all we
ean say is, that we are going to draw out of one of the
following, having no reason for supposing one in pre-
ference to anather > —

A BC DEF

LCs eo). 1 (aee) TH Goo «)}.. BV: 000)

A drawing takes place, and a white ball is produced,

consequently I. is immediately excluded; for from it

the observed event could not have been produced, This

much is certain; but we are also tempted to say that
E 3

54 ESSAY ON PROBABILITIES.

II. is rendered unlikely, because, from such an ante.
cedent state of things, a black ball would have been
more likely than a white one. On the same prin-
ciple III. is more likely than II., and IV. the most
likely of all. We have then to decide the relative pro-
babilities of IJ., III., and IV.

Before the drawing took place, the probability of each
set of circumstances was 4; and, the lottery being
given, the probability of any one ball in it was 1.
Thus the chance of III. being the lottery, and the
second white ball being drawn from it, was} x 1, or
jx The same of other balls: so that, in fact, our
primitive position was that of having to draw from 12
balls, 6 white and 6 black, all equally probable. But
the observed event changed that position ; a white ball
was drawn: was it a given ball (namely, the white ball
in II.), or was it one of two given balls (those in III.),
or was it one of three (those in IV.)? There are siv
cases in question, namely, A, B, C, D, E, F, and one
of them happened,— we do not know which. We have
used all the knowledge we have (namely, that a white
ball was drawn,) in excluding the black balls.

Hence the chance that A was drawn, or that

II. was the lottery - is 7:
That either B or C was Gran, or that Il.

was the lottery - = is 3%
That either D, E, or Fo was drawn, or that

IV. was the lottery - - - is 3

In the preceding instance, owing to the number of balls
being the same in every lottery, the antecedent proba-
bility of each ball was the same. Previous to deducing
a rule, I take an instance in which this is not the case.
Prosiem. A white ball has been drawn, and from
one or other of the two following urns:
(2 white, 5 black) (3 white, 1 black).

What are the probabilities in favour of each urn?

The case is not now that of a lottery of 5 white and
6 black balls; for the chance of our going to the first
urn (which is$), and thence drawing a given white

ON INVERSE PROBABILITIES. 55

ball (chance +), is 4 x +r =!;; while our chance of
going to the second urn (which is }), and thence draw-
ing a given white ball (chance }), is + x fori. But
since we do not alter the chance of producing a white
ball from either urn, if we double, or treble, &c. the
number of white balls, provided we at the same time
double, or treble, &c. the number of black balls, let us
put four times as many balls into the first, and seven
times as many into the second, as there are already.
Thus we have :

(8 white, 20 black) (21 white, 7 black).

There are now 28 balls ineach: every individual ball
has the antecedent probability x -=1,; and since our
knowledge of the event (a white ball was drawn) ex-
cludes the black balls, the question is simply this: —
Out of 29 possible, and equally probable cases, was
the event which did happen one out of a certain 8, or
one out ofthe remaining 21? ‘The chances of these
are 8, and 24; consequently it is 21 to 8 that the
second lottery was that which was drawn, and not the
first.

On looking at the resulting chance for the first urn,
namely, =, or the (21 + 8)th part of 8, we see that
8 and 21 are in proportion to the two chances for a
white ball being drawn, when we know that we are
drawing from the first urn, or from the second. For
these chances are 2 and 3, which, reduced to a common
denominator, are 8; and $4, which are in the propor-
tion of 8 to 21. The same reasoning may be applied
to any other cases, and the result is as follows : —

Principle V.— When an event has happened, and
the state of things under which it happened must have
been one out of the set A, B, C, D, &c., take the
different states for granted, one after the other, and
ascertain the probability that, such state existing, the
event which did happen would have happened. Divide
the probability thus deduced from A by the sum of the
probabilities deduced from all, and the result is the

E 4

56 ESSAY ON PROBABILITIES.

probability that A was the state which produced the
event: and similarly for the rest. [Or, reduce the re-
sults of the first part of the rule to a common deno-
minator, and use the numerators only in the second ©
part of the rule.]

Exampxte I. There is a lottery which is one or
other of the two following:

(3 white, 7 black) (all white). |

A ball is drawn, and restored; this takes place five
times, and the result is always a white ball. What are
the chances for each lottery?

Upon the supposition that the first lottery was that
in question, the chance of the observed event is the
product of 3°55 o> vo» To» and 35, Of tooo00: When
the second is the lottery, the observed event is certain,
and its probability is 1 or 190°8°0, Consequently, the
probability for the second lottery is 499009, or the
second has the odds 100000 to 243, or more than 411
to 1 in its favour.

Examp.ie II. Two witnesses, on each of whom it
is 3 to 1 that he speaks truth, unite in affirming that
an event did happen, which of itself is equally likely
to have happened or,not to have happened. What is
the probability that the event did happen?

The fact observed is the agreement of the two wit-
nesses in asserting the event: the two possible ante-
cedents (equally likely) are,—1. The event did happen.
2. It didnot happen. If it did happen, the probability
that both witnesses should state its happening is that
of their both telling the truth, which is? x #, or =%.
If it did not happen, then the probability that both
witnesses should assert its happening is that of their
both speaking falsely, which is + x 4, or51,. Conse-
quently, the probability that the event did happen is
the (9 + 1)th part of 9, or =% ; that is, it is 9 to 1
in favour of the event having happened.

Exampxe II]. There are two urns, having certainly ©
3 and 2 white balls; and in one or other, but which

ON INVERSE PROBABILITIES. 57

is not known, is a black ball. A ball is drawn and
replaced ; and this process is repeated, but whether out
of the same urn as before is not known. Both draw-
ings give a white ball: what is the probability of the
several cases from which this result might have hap-
pened ?

Since the black ball is as likely to be in one as in
the other, the antecedent state of things is (so far as a
single drawing is concerned,) the same as if there were
four urns, as follows:

I. (3 white) II. (3 white, 1 black) III. (2 white)
IV. (2 white, 1 black).

There are 16 possible cases, PP [2, 4,] numbered in
the first columns following, described in the second,
and having the probability which each would give
to the observed event (both drawings white) registered
in the third, together with the numerator, when all the
fractions are reduced to a common denominator 144.

PiLs 1144-8 GF ed 3, 108
Bick ied, Wa 61k eet
Sige le 60 Aen Pb? ca fy da ei a blo
4'T IV. | 4, 96 [8] ILIV: | &, 72
S(t (bei fy ae
10 | ILI. If. | 3;108 | 14] Iv. 11. | 45, 72
41. | TMI. IT. | 1,144 |. 15° | IV. IT & 96

12/ IIL Iv. |2, 96 | 16|IvV.Iv.| 4,64

That is to say, if (case 8), II. and IV. were the urns
of the first and second drawings, the chance of the
observed event is 4 or ,4°%. But, it must be remem-
bered, that we do not suppose the black ball may have
been removed from one urn into the other before the
second drawing takes place. Most of the preceding
cases are, therefore, to be rejected; in fact, I. can
combine with nothing but I. or IV., and II. with
nothing but II. or III. Reject, therefore, cases 2, 3,
5, 8, 9, 12, 14, 15, and the sum of the numerators in
the rest is 841. Hence the probability (for instance, )

:

58 ESSAY ON PROBABILITIES.

1. That the black ball is with the three white ones.
2. That the first drawing is from the lottery which has
the black ball, and the second from the other, is (case 7)
198, To find the total probability that the black ball
is with the three white ones, we must add the proba-
bilities of all the cases (as to ss rab ap hae can take
wee under this arrangement, namely #4,, 398, 298,
444, giving #41. Consequently, from the observed
event, it is slightly more probable that the black ball is
with the three white ones, than with the two.

The principle which we have illustrated, though a
mathematical consequence of those which precede, is
nevertheless received in common life upon its own
evidence. When an event happens, we immediately
look to that cause or antecedent which such event most
often follows. When it rains, we suspect the barometer
must have fallen; because, when the barometer falls,
it usually rains.

Our next step is to inquire, what is the probability
which an event gives to its several possible antece-
dents, upon the supposition that they are not all equally
likely beforehand ; as in the following instance.

Prosiem. A white ball is drawn, and from one or
other of the following urns:

(3 white, 4 black) (2 white, 7 black):

but before the drawing was made, it was three to one
that the drawer should go to the first urn, and not to
the second. What is the chance that it was the first
urn from which the drawing was made P

We may immediately reduce the preceding to the
case where all the antecedent circumstances are equally
probable, by introducing urns enough of the first kind
to make it 3 to 1 that the drawing is made from one
or other of them. Let us suppose the urns to be as
follows :
(3 white, 4 black) (3 white, 4 black) (3 white, 4 black)

(2 white, 7 black):

these urns being equally probable, the hypothesis of

ON INVERSE PROBABILITIES. 59

the problem exists. If we number the urns A,, A,,
A, B, the chances which they severally give to the
observed event are #, +, 4, and %, the numerators of
which, reduced to a common denehiion diel, ae: 27,27,
27, and 14, Consequently, the probability that A , was
chosen, is 34; and the same for A, and A,. There-
fore, the chance that one a other of the three, A,, A,,
and A., was chosen, is 84; which is the probability of
the ball having been drawn from the urn (3 white,
4 black,) in the first statement of the problem.

The rule to which the preceding reasoning conducts
us is as follows: When the different states under which
an event may have happened are not equally likely to
have existed, then having found the probability which
each state would give to the observed event, multiply
each by the probability of the state itself before using
the rule in page 55. The following is another example.

An event has happened, the possible preceding states
of which are represented by A, B, and C. The chances
of the existence of these different states (independently
of all knowledge of the observed event) are, say, 4, %,
and : the probabilities that the observed event would
have happened are -’,, =4;, and =, if A, B, or C were
certainly existing. Form the three products
Probability that the event rie} he.

Probability of A x { happen if A were known to exist

there are *

4 5 3 4 2 Hae
+X pp $ X pp and 5 x +73

- the numerators of which (the denominators being com-
mon) are 20, 12, and 4. Then the probability that
A was the state under which the event happened, is
20 divided by 20 + 12 + 4, or 2 3 ; those of B and C are
z¢ and 4

Let us now suppose, that “havihig only a first event
by which to judge of the preceding state of things, we
ask what is the probability of a second event yet to
come. For instance, an urn contains two balls, but
whether white or black is not known; the first draw-

60 ESSAY ON PROBABILITIES;

ing gives a white ball, and the ball is replaced. What
is the chance that a second drawing shall give a black
ball P

The preceding states under which the first event may
have happened, are —

(2 white) (1 white, 1 black);

and 1 and 4 are the chances of a white ball, if one or
other state were absolutely known to exist. Hence, by
the last principle, { and + are the chances which the
observed event gives to the two states; that is, it is
two to one that both balls were white. Now, the black
ball can only appear at the second drawing, upon the
supposition of the second state existing; and this sup-
position being made, the chance of a black ball at the
second drawing is 4. Hence, page 43., the second
event depending upon two contingencies, of which the
chances are + and 4, its chance is 4, or it is five to
one against the second drawing being black. But let
us now ask what is the chance of a white ball at the
second drawing? LEither of the preceding states admit
of such an event, and, in fact, the event proposed — a
white ball at the second drawing — means
One or other of these i (2 white) and white drawn.

two combinations. ( (1 white, 1 black) and white drawn, ©
In the first combination, the first contingency (the
chance of which is 1) ensures the second: so that
+ x 1 is the chance of a white ball being drawn, and
of (2 white) being the lottery from which it is drawn.
In the second combination, the chances of the two con-
tingencies are } and 1, whence } is the chance of a
white ball being drawn, and being drawn from (1
white, 1 black). But the event proposed i if
either of these cases occur; therefore, 2 + 4, or 2, is
the chance of a white ball ‘at the second drawing, as
might have been inferred from the probability already
obtained for a black ball. By such reasoning as the
preceding, the following principle is established :

Principle VI. Having given an observed event A,

ON INVERSE PROBABILITIES. 61

to find the probability which it affords to the suppo-
sition that a coming event shall be B, find the proba-
bility which A gives to every possible preceding state ;
multiply each probability thus obtained by the chance
which B would have from that state, and add the
results together.

Propiem. There is a lottery of 10 balls, each one
white or black, but which is not known: drawings are
made, after each of which the ball is replaced. 'The
first five drawings are white; what chance is there
that the next two drawings shall be white ?

Let S (20) denote the sum of all numbers up to 20 ;
S (202) the sum of the squares of all numbers up to
202 or 400; and soon. The possible preceding states
are —

(1W,9B) (2W,8B.)....(10 W,0B);
and the probabilities of W five times running from
each, are

To ioedesto+ds Tot rot rot rote &.
up to 49.40.49 .10 . 49, or 1, the event being
certain, if the last state existed, The numerators of
these products (the common denominator being 10°,)
are 15, 25, cree 105: whence, page 55., the proba-
bilities of the several states are —

15 25 105

Sic’ Sie” .. .810
By the same reasoning, the probabilities of the proposed
events (two more white balls,) are —
SS Se ene 102

102102 102 ;
the different preceding states being successively sup-

posed to exist; whence the actual chance which the
observed event gives to the proposed is —

15 12 25 2 5 2
a agora x —_— —_—_—— x 2 + eceese + econ x te"
S105 102 § 105 102 $105 1023
ree S 107
which is

102 § 105,

62 ESSAY ON PROBABILITIES.

By precisely the same reasoning, if there had been
1000 balls in the lottery, and if 157 had been drawn
white, the probability that 27 more drawings would
have given white balls, would have been

S 1000184
(157 + 27 = 184) 100027. S$ 1000157

The difficulty of calculating S 1000!54 is insuperable :
but a mathematical theorem which we shall proceed to
explain, makes it very easy to find a near approxima-
tion to the preceding result, and the nearer the greater
the numbers in question. :

Take the sums of the powers of the different num-
bers, as follows :

Firstpowers 14+ 2+ S+ 44+ 5 GH sseoee
Squares 1+ 4+ 9+ 164 25+ 36+ cence,
Cubes 1+ 8+ 27+ 64+ 125+ 216 +...

Fourth powers 14+ 164 81+ 256+ 62541296 + sess
Fifth powers 1432 +243 4+102443125 4+ 7776 + cess.

a mm] mp] ogee be] re

and examine the sum of any number of terms in any
line, as compared with the term immediately below the
last in the sum; thus : —

14+24+3+4+4 1+164+81+4+256 +625

16 3125

Form fractions with such sums as numerators, and their
compared terms as denominators, and observe how
much each fraction, so formed, differs from the fraction
written in the last column, as follows: —

14+2 3 see 142+3 6 1 1
a ee geese ores Fae! he pe

4 4.2 4 9 Se ae

eee ead ees 18 eh aot on;
46-16, 2: 8 25. 25 2 10

hanes S (any number) 1 1

square of that number 2” twice that number

Hence it follows, that when the number is large, the
preceding fraction is very nearly one half, or 1 + 2 +
3 + .. up to a large number, is very nearly one half
the square of that number.

ON INVERSE PROBABILITIES. | 63

; 044 34°02 14449 1 £
Again — ——— Se Se Ht = =
e394 27 3 27
S 42 $0 °3. 418 S 52 | ice
6horae 64S BE i2sor58 3. 75

B10? (S850: Bb 485

ak ce te ; and so on.
103 1000 3. 8000

In this way it appears that the sum of all the
squares of numbers is nearly one third of the cube of
the last number, and that the greater the number of
squares taken, the greater the proximity in question.
This proposition is general, namely, that the sum of the
nth powers of numbers is nearly the (n + 1)th part
of the (n + 1)th power of the last of the numbers :
thus, the sum of all the 13th powers 1/85 + 218 4.
up to 100015, is very nearly the 14th part of 100014.
This proposition, never absolutely true, may be made as
near the truth as we please, by taking the number of
terms sufficiently great; and the error made by the
substitution, is nearly such a fraction of the whole as
has one more than the index of the power for its nu-
merator, and twice the number stopped at for its
denominator. Thus, if the tenth powers of all num-
bers were summed up to 10,0001°, the substitute for
this sum given by the theorem, namely, =, of 10,000!!,
would be too small by about

oT eee
2x10,000 20,000

of the whole.

Prosiem. A lottery contains 10,060 balls, each of
which may be white or black. A ball is drawn and
then replaced, and 100 such drawings give nothing but
white balls: what is the chance that the five next
drawings shall all be white ?

S 10,000100+5
10,0005 S 10,000100

This chance, by what precedes, is

But § 10,000100+5 mie J x 10,000106 very nearly .
106

64 ESSAY ON PROBABILITIES.

S 10,000100__! 4 10,000!91,,,...006
101

Consequently, the chance as TOE of 10,0001" or abe nearly.
sor of 10, 000106 106
If the number of balls had been a million, instead
of 10,000, the preceding odds, namely, 101 to 5,
would still more nearly have represented the chance
that after 100 drawings, all white, the next five should
be white also. If the number of balls had been abso-
lutely unlimited, the preceding odds will correctly ex-
press that same chance. But a lottery with an unlimited
number of balls, each of which may be either white or
black, is a lottery which may be anything whatever.
For instance: [3 W, 7 B] is equivalent to an unlimited
lottery, in which for every 7 black balls there are 3
white ones. Again, an unlimited lottery in which any
number of balls may be black, and the rest white, is
one in which the chance of drawing a white ball may
be any whatever, and is absolutely unknown. The draw-
ing of a white ball from such a lottery may be likened
to the occurrence of an event, about the preceding
chances of which we are in total ignorance. The pre-
ceding process furnishes us with the following theorem.
When an event which may, for any thing we can see
to the contrary beforehand, happen in either of two
different ways, happens one way m times in succession,
it ism + 1 to n that it shall happen » times more in
the same way, if it happen n times more at all. Thus,
suppose a person on the bank of a river, not knowing
in what country he is, and not having the smallest
reason to know whether the vessels which come up the
river carry flags or not: the first ten ships which
come up all carry flags (m = 10); thenitis 10 4+ 1
to 3, or 11 to 3 that the next three ships shall carry
them, and 10 + 1 to 1, or 11 to 1 that the next ship
shall carry a flag. And it is always m + 1 to 1, that
an event which has occurred in one out of two possible
Ways m times in succession, shall happen the same way
on the next occasion.

ON INVERSE PROBABILITIES. 65

The preceding affords some view of the way in
which chances are obtained, in cases where the ante-
cedent probability of the events stated may be any
whatever. The following are conclusions upon the same
subject, obtained by a more complicated reasoning of
the same kind.

If an event, each repetition of which may be either
A or B, have happened m+ 7 times, and if A have
occurred m times, and B n times; then it is m+1
to m+1 that the next event shall produce A, and
not B. And in the same case the chance that out of
p+4q events to come, p shall produce A, and q shall pro-
duce B, is (see pages 15 and 16, for explanation of [ ]).

[p+q] x[m+1,m+p] x [nm+]1,n+q] |

[p] x [q] x [m+n+2,m4+n+p+q4+1]

[m+1,m+p] aed [z+1,2+9]
[m+n+2,m+n+pt+1} [m+n+2,m+n+q+1]
are the chances that in p new events, all shall give A.
and that in q new events all shall give B.

Exampire. In a lottery containing an unlimited
number of balls, in which the proportion of black and
white is absolutely unknown, six drawings give four
white and two black; what are the chances that four
drawings more shall give all white, or one only black,
or two only black, &c.

Let us first take the case of three white and one black :
here ot = 4, n= %, 9.4 3, ¢ = 1,

[p+q] = 1.2.3.4, [m+1,m+p] = 5.6.7,
In+l,n+g] =3[p] = 1.2.3. [g] =1,
[m+n+2,m+n+p+q+1] = 8. 9.10.11.
1.2.3.4. % 5.6.7. x 3. 7
1.2.3. x 1.x 8.9.10.11. 99
That two shall be white and two black (m = 4, n = 2,
p = 2, q = 2), the chance is
1 2.3.4, x 5.6.x 3.4. 9 3
1.2. %1.2.%8.9.10.11. 11
That one shall be white and three black (p = J,
q = 3), the chance is Pe

and

Chance required is

F

66 ESSAY ON PROBABILITIES.

1.2.5,4%5%8.4.5 9.5
1x 1-2.3x8.9.10.11 33

| That all shall be white, or all black (p = 4,¢ = 4,
second and third formule), the chances are

5.6.7.8 , or irs 3.4.5.6 pe 1

§.910.11 33 89.10.11 22

and the verification of the whole is

7 3 em |

a3 *11* 33" 33* a2 —

which must be, since one or other of the cases con-
sidered must happen.

When it is known beforehand that either A or B
must happen, and out of m + nm times A has hap-
pened m times, and B n times, then (page 65.) it is
m+1 to n+1 that A will happen the next time.
But suppose we have no reason, except what we gather
from the observed event, to know that A or B must
happen ; that is, suppose C or D, or E, &c. might have
happened: then the next event may be either A or
B, or a new species, of which it can be found that the
respective probabilities are proportional to m + 1,
m+1,and 1; so that though the odds remain m+1
ton +1 for A rather than B, yet it is now m+ 1
to n+ 2 for A against either B or the new event.
Thus, suppose a game at which one party or the other
must win, and suppose that out of 20 games A has
won 13 and B 7: and this is all we know of the game
or of the players, Then, it is 134+1 to 7+1, or
14 to 8, or 7 to 4, that A shall win the 21st game.
But suppose that it is possible to have a drawn game;
then there is some chance that the 2lst may be a
drawn game, though but a small one, as might be
inferred from such a thing never happening in 20
trials. The 21st game may be either A’s or B’s, or
drawn: of which the chances are as 1341, 7+],
and 1; or as 14, 8, and 1. Consequently, though in
the preceding case it was 14 out of 22 in favour of A’s

ON INVERSE PROBABILITIES. G7

winning, it is now 14 out of 23, and 1 chance out of
23 remains for the next game being drawn.

When a number of different events have happened,
A, B, C, &c., write down each number increased by 1,
and the results will express the several relative proba-
bilities, on the supposition that no events can happen
except those which have happened. But if new events
may happen, write down 1 for the relative probability
of such an occurrence at the next trial. Thus, if out
of a box, and in 100 drawings, there have appeared 49
white balls, 377 red, and 14 black ; then if it be known
that nothing but white, red, or black can appear, con-
sider the chances of these to be as 50, 38, and 15; that
is, °°; is the chance of drawing a white ball at the
101st trial. But if another sort of ball may appear,
then the chances of the four cases being as 50, 38, 15,
and 1, it follows that =°°. is the chance of a ie ball
at the 101st trial.

In judging of future events by those which have
passed, we must be extremely cautious always to pre-
serve the same method of considering the event pro-
posed. If, for instance, in 100 trials, A has appeared
49 times, B 37 times, and C 14 times, we know that
there is one chance out of 104 that the 101st drawing
shall give neither A, B, nor C, but something else.
What the new character may possibly be is left un-
known ; it may be another letter, or it may be a num-
ber, 2 picture, or a blank. Are we to understand that
all these are equally probable? Common sense tells us
the contrary ; experience makes us feel it much more
likely that the letter D should appear at the 101st trial,
than any stated number or picture. But we have
now changed the question, and, dropping the distinction
between A, B, and C, have considered them merely as
letters. Having drawn a letter 100 times running, we
are to infer (page 64.) that it is 101 to 1 in favour
of our drawing a letter at the 101st trial; or that 1°
is the chance of this. But it is already 103 to 1 that
the next drawing shall be, not merely a letter, but one

F 2

68 ESSAY ON PROBABILITIES.

of the letters A, B, and C: that is, to all appearance,
we have this strange result ; — the chances of drawing
one of the three, A, B or C, are greater than those of
drawing one of the set, A or B, or C or D, &c. &c. up
to Z. This paradox will afford me a good opportunity
of again inculcating the maxim, that the probability of
an event is the presumption drawn from certain obvious
principles, as to what the state of our minds ought
to be with regard to belief in the happening of that
event, as influenced by our knowledge of previous
events. Consequently, if John know that A, B, and
C have been drawn 49, 37, and 14 times, and nothing
more, he has reasonable ground, with his knowledge,
for assenting to the proposition that the 101st trial
shall give either A, B, or C, as to a proposition
which has 1°93 of probability, or 103 to 1 in its
favour. But if Thomas only know that 100 draw-
ings have all given letters, then he, with his know-
ledge, has no ground of inference with respect to A, B,
and C, in particular, but may reasonably assent to the
proposition, “ the 101st drawing will also give a letter,”
as having a probability of 4°1, or 101 to 1 in its
favour. But the paradox in question requires that
John should make believe he knows no more than
Thomas, and then be surprised that a discordance
should arise from his using that knowledge in recon-
sidering a result, which he has suppressed in the me-
thod of attaining it.

It very rarely happens that we meet with a case in
which we can so distinctly specify the antecedent cir-
cumstances which infiuence our assent or dissent, as to
enable us to apply direct calculation to the determin-
ation of the rational probability of an event to come.
And in most of the practicable cases, the largeness of
the numbers employed would check our progress, if it
were not for the approximative methods of the higher
mathematics, and the tables at the end of this work, I
now proceed to the method of using those tables.

USE OF TABLES. 69

CHAPTER IV.

USE OF THE TABLES AT THE END OF THIS WORK.

I nave endeavoured to accumulate in this chapter
a considerable part of the uninteresting details of com-
putation which accompany the solution of complicated
problems. It is at the readers’ pleasure to omit the
whole of it, referring to it afterwards in cases where
its assistance may be necessary.

In Table I. we see — (I.) a column headed t, con-
taining the series -00, °O1 ..... "O09, TU casaw oe
every hundredth of a unit from 0 to 2 —(II.) a column
headed H, deduced in the manner pointed out in page
17,—(iII.) columns headed A and A®%, which are
only the differences of the numbers in column H
(marked- A), and the differences of those differences,
(marked A’). The following is an extract from the
table: —

t H A A2
‘47 | 49374 52} 900 46] 8 61
48 | 50274 98 | 891 85 | we. P
“£9 “SESGGi SSF ccc covccs F veces ‘

The columns A and A? must be made up to 7 places
of decimals * by means of ciphers: thus, 90046 means
-0090046 ; and 861 means -0000861. The formation
of A and A is then as follows: —

-5027498 “5116683 0090046
4937452 5027498 0089185
Subt. 0090046 0089185 0000861

* A little practice will show how to dispense with the decimal points
altogether till the end of the process.

ge 3

70 ESSAY ON PROBABILITIES.

Since it is very seldom necessary to use more than five
places of the table marked t, the sixth and seventh places,
and those which arise from them in the differences, are
separated from the rest by a blank space. The sixth
and seventh places are allowed to remain, on account of
the use which will hereafter be made of the differences
derived from them.

In Table IJ. we find to five places only a column
marked K, and another marked A, containing the dif-
ferences of the former column. This is a modification
of the former table, the reason of which will hereafter
appear. In the meanwhile, however, observe that we
can directly find the value of H and K_ by these tables
only, when t is ‘00 or ‘01, &c.; that is, when t is a
given decimal of two places. But supposing it required
to find H when t lies between two of the values in the
table ; suppose, for instance, we ask what is H when

= ‘47694? The method is as follows:

Question. What is the value of H (Table I.)
when t = °4'7694, correct to five places of decimals ?

Rute. / EXeEMr.iricaTION.
Take out of Table I. the Opposite to +47 we find
value of H answering to the 49375 *

two first decimal places and
the whole number preceding
them, if there be one. Retain
only five places of decimals.
Take the figures of the first | Three figures remaining, 694,

difference (as far as the blank 900 x 694=62460U.
space), and multiply them by Cut away three = Aeuree,
the remaining figures in the , (685.2

value of t, and cut away as
many places from the result as
there were remaining figures.

Add the figures in the last 49375
result to the right hand of the 625
first, and the sum is the an-
swer required, - 50000

When t= °47694, H = °50000.

* Whenever decimal figures are rejected, if the first rejected be five or
upwards, the last retained is increased by a ‘unit.

USE OF TABLES. ye |

As another example, suppose the value of t to be

1°51209.

When t=1°51, H=.96728
24

When t=1°51209, H=*96752

A=114

209

1026
228

23,826

We shall now take the inverse problem, and sup-
posing H to be intermediate between two values in the

table, require the value of t.
*93972.
Rute.
Find in the table the value

of H next below the given | n

value ; note the corresponding
value of t, and subtract the
nearest value of H from the
given value.

Annex three ciphers to the
difference just found, and di-
vide by the figures of the dif-
ference in the table which
come before the blank space,
rejecting fractions, and taking
the nearest whole number.

The quotient cannot have
more than three places: if
three, annex to the value of t
already found; if less than
three, place ciphers at the be-
ginning to make up the defi-
ciency, and annex.

For instance, let H =

EX£EMPLIFICATION.
H *93972,

ext below
athe aie} 93807t= 1°32

165
Tabular diff. 195.

195)165000(846
1560

900
780
1200
1170

30

t=1°32 846

72 USE OF TABLES AT END OF WORK.

Let H = *97169; required the value of t?

H= *97169
Nearest below =°97162 ceccocseesee t= 1°55

101 )7000(69 t=1°55069 Answer.
606

940
909

ST arama

31 +

The Table II. is used in exactly the same way, except
towards the end, from t = 3°40 upwards; in which
case the cipher at the end of t must be neglected, and only
one decimal place taken out of the table in the value
of t. For instance, to find the value of t answering to
K= "98222,

K = *98222
Next below °98176 evccceccoses t=3°5

806 )46000( 150 t==-3°5150
306
1540
1530

100

In the first table, there is another result which will
frequently be wanted, and which I shall call H’% It
arises from adding half the second difference to the first
difference *, if the value of t be in the tables, and
making five decimal places. But if the value of t be
not in the tables, then H’ must be formed for the
values of t immediately above and below ; and by means —
of the first and the difference of the two, H’ must be
found in exactly the same manner as H is found in
the first of the preceding rules, page 70., remembering
to subtract at the last step instead of adding, if the second
H’ thus previously determined be less than the first.

* Meaning the whole differences ; not the parts which precede the blank
space, as in the preceding rules. |

—~I
Q9

USE OF TAELES.

Examete 1. When t is 1°56, what.is H’?

A= 9745
t= 151
H’ = :09896 :
Exampie 2... When t — 1:23412, what is H’?
t= 1°23 H’ = :24852 :
t=1:24 H’=-24245 Diff. 607
412 "24852
607 250
2884 Subt. -24602=H’ when t=1°23412
2472
250,084

_ We have already seen that when two events, A and
B, one of which must happen at every trial, have
severally happened m times and n times in m+ n
trials, it is m+ 1 to n+ 1 that A shall happen
at the next trial. But m + 1 to n + 1 is very
nearly m to n, when m and n are considerable num-
bers: for instance, 248 to 117 is very nearly 247 to
116. That is, when a great many trials have been
made, the numbers of times which A and B have hap-
pened express very nearly the odds (relative proba-
bilities) for A against B; or, inverted, for B aguinst A.
Let us convert the problem, and supposing that we
know beforehand the chances of A and B, are we to
suppose that in agreat many trials A and B will happen
in proportion to their respective probabilities P Common
sense tells us that such will always be nearly the case,
but that the odds are great against an exact result,
Suppose 3000 drawings to be made from a lottery con-
taining two As and one B. We must then, it seems
clear, draw A twice as often as B, in the long run.
Our reason convinces us thus. Let one of the As be
distinguished from the other by an accent, so that we
have A, A’, and B. If the urn be well shaken before
each drawing, it is impossible to believe that, in the

ESSAY ON PROBABILITIES.

74

whole result of 3000 trials, we shall have drawn the
three in very unequal numbers ; so that, destroying the
distinction between A and A’, we feel secure of drawing
A twice as often as B; and it is obviously two to one

in favour of A at each trial.

The following phrases

- seem to common sense to mean the same thing.

It is two to one that A
shall happen, and not B.

It is an even chance for
head or tail.

It is more than a hundred
to one, that a ship at sea will
not be lost.

In the long run, A will
happen twice as often as B.

In a large number of tosses,
the heads and tails will occur
in nearly equal numbers,

Of all the ships which sail,
the number which is not lost
exceeds that which is lost more

than a hundred times.

I now proceed to some problems, which will exhibit
the method of applying the tables, and will illustrate
and confirm the preceding notions.

Prosiem I. The odds for A against B being a to 3,
to find the chance that in n times a + 5 trials, A shall
happen exactly n xa times, and B n x b times.

Rue. Divide the H’ belonging to ¢ = 0 (page 72)
by the square root of the following: 8 times the product
of n, a, and b, divided by a + b.

Suppose, for instance, a die is thrown 6000 times ;
what is the chance that exactly 1000 of the throws shall
give an ace? Here it is 1 to 5 that an ace shall be
thrown in any one trial, and 6000 is 1000 times 1+ 5.
Hence a = 1,b = 5, n = 1000: 8 times the product
of n, a, and b is 40,000, the sixth part of which is 6667
(sufficiently near), and the square root of this is 81°65.
Again, when ¢ = 0, we have in Table I.

A=112833, | A2=11; whence H’ is 1*12844

and 112814 divided by 81:65 gives ‘014 very nearly,
This is very near the real probability that 6000 throws
with a die shall give exactly 1000 aces: for such an
event there are only 14 chances out of a thousand ; and

USE OF TABLES. 75

it is 1000 — 14 to 14, or about 704 to 1, against the
event. This result is rather above what we should have
expected ; we might have imagined it to be more
than 71 to 1 against 6000 throws giving exactly 1000
aces.

As another example, let us find the probability that,
out of 200 tosses with a halfpenny, there shall be ex-
actly 100 heads and 100 tails Herea=1,b=1,
nm = 100, and 8nab* is 800, which divided by a + 3,
or 2, gives 400, the square root of which is20. And
H’, when ¢=0 (or 1'12844) divided by 20, gives :056.
It is therefore about 944 to 56, or 17 to 1, against the
proposed event; and (page 42.) we must repeat 200
throws 12 times to have an even chance of the equality
of heads and tails happening once.

Generally speaking, the rules in this chapter are very
accurate only when the number of trials is considerable.
Suppose only 12 tosses; required the chances of 6
heads and. 6 tails. Here. a = 1, b= 1, n = 6;
8nab = 48, which divided by 2 gives 24, whose square
root is 4°9 very nearly. And 1°12844 divided by 4°9
gives °23, or 77 to 23, that is 3.8, to 1 against the
event. That is (page 47), this rule is not very inaccu-
rate, even when the number of trials is as low as 12.

We shall call the event whose chance is sought in the
preceding problem, the probable mean; understanding
by that term the event which is more likely to happen
than any other. Thus, when 12 halfpence are thrown
up, 6 heads and 6 tails is the probable mean, being the
event which is more likely than any other, though not
in itself more likely than not. When 6000 throws are
made with a die, the probable mean is 1000 aces, 1000
deuces, &c.

ProsiEM II. The odds for A against B being a to },
required the chance that in n times a + 0 trials, the
As shall fall short of the probable mean by a given

* Juxtaposition of numbers, in algebra, stands for their product.

"6 ESSAY ON PROBABILITIES.

number /: J being small, compared with the whole number
of As in the probable mean.

Rute. Divide twice / by the square root obtained in
the last example ; and find the value of H’ from Table I.,
taking the preceding quotient for t. Divide H’, so found
by the square root just used, and the quotient is the
answer required.

N.B. This rule also applies when the number of As

is to exceed the probable mean by 1.
_ Examrte I. In 6000 throws with a die, what is
the chance that the aces shall fall short of (or exceed)
1000 by exactly 50°? Herea = 1, 6b = 5, n= 1000,
and the square root is 81°65, as before. And twice /,
or 100, divided by 81°65, gives 1°22: to which the
value of H’ is 25162 + 4 of 611, with five decimal
places, or ‘25468. This last, divided by 81°65, gives
"0031 ; so that it is about 997 to 3, or 332 to 1, against
the proposed event: and the 6000 throws must be re-
peated 232 times to give an even chance of succeeding
once.

ExamptE II. What is the chance that in 200 tosses,
there shall be exactly 95 heads? Herea=1,b)= 1,
nm = 100, / = 5, and the square root, as before, is 20.
And twice J, or 10, divided by 20, gives °50, which
being t, the value of H’ is ‘87882, which divided by
20 gives ‘044 very nearly. It is, then, 956 to 44, or
about 22 to 1, against the proposed event.

Exampxe III. In 12 tosses, what is the chance of
exactly 7 heads? Here a=1, b=1, n=6,/=1,
the square root, as before, is 4°9, and 2 divided by 49
is ‘41 nearly ; which being t, H!' is 95384, which
divided by 4°9 gives *194. It is therefore 806 to 194,
or 4.°, to 1, against the proposed event. In page 47.
it is 3304 to 792 against this event, or 4} nearly. Hence
the incorrectness of our rule is very small.

Prosiem III. The odds for A against B being
a to b, required the chance that in n times a + b throws,
the number of As shall not differ from the probable
mean by more than /, |

USE OF TABLES. TZ

Rute. Divide one more than twice / by the square
root already mentioned, and the quotient being made
t, the value of H in Table I. is the probability re-
quired.

Exampte I. In 6000 throws with a die, what is
the chance that the number of aces shall not differ from
1000 by more than 50; that is, shall lie between 950
and 1050, both inclusive. Here a=1, b= 5,
nm =1000, /= 50, and the square root as before is 81°65.
Divide 27+ 1, or 101, by 81°65, which gives 1:237,
which being t, H is 91977. Hence it is 920 to 80 in
favour of the proposed event, or about 114 to 1.

Exampte II. In 200 tosses, what is the chance that
the number of heads shall lie between 97 and 103, both
incisive? Here a=: 1, b= 1,7 — 100, }= 3: age
the square root, as before, is 20. And 2/-+ 1, or 7,
divided by 20, gives *35, which being t, H is °379.
Hence it is 621 to 379, or about 31 to 19, against the
proposed event.

Exampte III. In 12 tosses, what is the chance of
the heads being either 5, 6, or 7 in number? Here
a=1, b=1, n= 6, i=1, and the square root, as
before, is 4°9. And 2/ + 1, or 3, divided by 4°9, gives
61, which being t, H is 6117. Hence it is about 612
to 388, or 153 to 97, in favour of the proposed event,
In page 47 the chance of this event is

792 +92944792 : 2508
T
4096 4096

or *612

ProstemIV. The odds for A against B being a to,
and n times a + 6 trials being to be made, for what
number is there a given probability H that the As shall
not differ from the probable mean by more than that
number ?

Rue. Find in Table I. the value of t answering to
that of H (page 71), multiply it by the square root
alréady described, subtract 1, and divide by 2: the

78 ESSAY ON PROBABILITIES.

quotient, or its nearest whole number, is the answer
required.

Exampite. In 6000 throws with a die, within
what limits is it two to one that the aces shall be con-
tained? The square root is 81:65, and H is 2 or ‘66667,
to which the value of t is ‘68409, found as follows
(page 71) : —

H = *66667
Nearest below ‘66378 t=°68

Tab. Diff. 706) 289000(409 t= °68409 say °6841
2824 81°65

6600 34205
6354 41046
6841
246 54728

55°856765
] e

2)54+86

27°43 =1.

Answer. It is a little more than 2 to 1, that the
aces shall lie between 1000 — 28 and 1000 + 28, and
a little less than 2 to 1 that they shall lie between
1000 — 27 and 1000 + 27.

But the most convenient way of solving this problem
is by first finding for what degree of departure from the
probable mean there is an even chance. In this case,
since H =°5 (page’70), t is = °4'76936, which the
method in page 41, will show to be very nearly 24. It
will be worth while to re-state the whole process.

The odds for A against B being a to b, and the pro-
posed number of trials being n times a + 6, required
the limits of departure from the probable mean na,
within which it is an even chance that the number of
As shall be contained.

Ruiz. Multiply together 8,n,a, and b, and divide by

USE OF TABLES. 79

a + b: extraet the square root of the quotient, and
multiply it by 31: subtract 65, and divide by 130: the
nearest whole number is the answer required. Thus in
the preceding instance, where the square root is 81°65,
multiply this by 31, which gives 2531-15 ; and 65 less
is 2466°15, which divided by 130 gives 18°97. Hence
it is very little less than an even chance that the aces
in 6000 throws shall be between 1000 + 19 and
1000 — 19, or 1019 and 981.

Having found the limits of departure for which there
is an even chance, we can now use Table II. as follows.
The values of t in Table II. are the proportions of
various departures (each increased by °5) to that depar-
ture which has an even chance, as just ascertained, and
also increased by 5: the values of K are the proba-
bilities of the departures answering to those of t. Having
then ascertained 18°97 to be the departure for which
there is an even chance, suppose I ask what is that limit

of departure within which it is two to one that the
aces shall be contained. Two to one gives 2 for the
chance, or ‘66667 : I look into Table II., and find that
when K is 66667, t is 1°43433, found as follows: —
K= ‘66667 t=1°43
Next below 66521

Tab. Diff. 337) 146000(433 t=1°43433
1348

1120
1011

1090
1011
79

This is the proportion which the departure in ques-
tion, increased by °5, bears to 18°97 increased by °5 or
19°47. Multiply 1°43433 by 19°47, giving 27:93 ;
from which subtract °5, giving 27°43 for the limit of
departure, the same as in page 78.

80 ESSAY ON PROBABILITIES.

Suppose the question to be that of page 77, namely,
what is the probability that the number of aces in 6000
throws shall lie within 50, one way or the other, of the
probable mean 1000? Now, 18°97 + °5 is 19°47, and
50 + °5 is 50°5, and 50°5 divided by 19°47 gives
2°594, which being t (in Table II.), K is -91981, ex-
tremely near to the result in page 77.

There are, therefore, two distinct methods of treating
these problems, connected with the two tables: and this
is a great advantage, since it is a very strong presump-
tion of a correct answer, when the results of the tables
agree. The problems III. and IV. being of great import-
ance, I shall now recapitulate their details, with the ad-
dition of some new phraseology. Let the instance be
6000 throws of a die, and the event A the arrival of an
ace. and B the arrival of some other face. The most
probable number of aces is 1000, though the arrival of
that exact number is not probable in the common sense
of the word. There will then most likely be a de-
parture from the number 1000 in the number of aces
thrown ; of which departure we are now entitled to
say, that it is very improbable it should be considerable.
Let the term neutral departure mean that degree of de-
parture for which it is just an even chance that the
actual event shall be contained within its limits: in
the present instance it is 18°07. We may explain the
fraction as follows: suppose a person to receive 1001.
for every unit by which the number of aces falls short
of or exceeds 1000. Then, supposing him to try this
stake a great many times, he will in the long run re-
ceive less than 1897/. at a trial, as often as he receives
more. But his receipts will oftener exceed than fall
short of 18002. ; while they will oftener fall short of
than exceed 1900/. Roughly speaking, there is here the
same probability that the aces shall not lie between
1000—19 and 1000+ 19 (both inclusive), and that
they shall lie between these numbers.

In all these problems there is a square root to be
found, which we call the square root, as there is no other

8

The odds are a to b, for A against B, and n (a + 5)
trials are contemplated. Though we have only instanced
whole values of n, yet it may be a fraction: thus, if the
odds are 3 to 2, and 96 trials are contemplated, n (3 + 2)
must be 96, or n must be 191. In this case, the pro-
bable mean is that A shall happen 573, and B 382; by
which it must be understood, that a person who should
repeat 96 throws a great many times, receiving 1/. for
every A, would, in the long run, gain on the average
573. per trial of 96 throws.

The square root in question, represented algebrai-

cally, is

USE OF TABLES.

eee

8nab
a+b

or the square root of the product of 8, n, a, and 6 di-
vided by a+ 8. I now subjoin the two principal pro-
-blems, with the two rules in parallel columns.
Prosuem. What is the chance~that the number of
As in n(a+6) trials shall lie between na +/ and
na—Jl, both inclusive? or what is the chance that the
departure from the probable mean shall not exceed / ?

By Taste I. By Taste II.

Find the square root, and
divide one more than twice /
by it; call the result t, and
find H answering to t in the
table. (Use the rule in p, 70.
if necessary.) This H is the
probability required.

Prosiem. What is that

Find the square root, and
multiply it by 31; then divide
by 180. To 7 add ‘5, and
divide by the preceding
quotient; call the result t,
and find the value of K
answering to t: this is the
probability required.

N. B. The neutral departure
is *5 less than the quotient first
found.

degree of departure within

which it is p to qg that the number of As in (a + 6)

trials shall lie ?

G

ESSAY ON

By Taste I.

Divide p by p +4, and call-
ing the result H, find the cor-
responding value of t in the
table. Multiply it by the
Square root, ‘subtract 1 and
divide by 2; the quotient
being called /, it is then p to q
that the As in 2 (a+b) trials
shall be contained between
na—l and na+l, both inelu-
sive.

PROBABILITIES.

By Tazsvez Il.

Find the square root ; mul-
tiply by 31, and divide by
130. Divide p by p+q, and
calling the quotient K, find the
corresponding value of t in
the table; multiply t by the
preceding quotient, subtract °5
from the product, and J being
the remainder, it is then p to q
that the Asin n (a+b) trials
shall be contained between
na —l and na + 1, both inclusive.

I shall conclude this problem with an example of each

ease, worked by both methods, without explanation.
There is a lottery containing 3 white and 2 black balls :
what is the chance that in 50,000 drawings the num-
ber of white balls shall be between 30,000 + 100 and
30,000- 100°

a =3 b=2, n=10,000, I= 100 809'84
8 x 3x 2x 10,000 a
5 = 96,000 | 130)9605-04(73°885
56,000 = 309-84 73°885)100°5(1°3602=t
309-84) 201(-64873 =t K = 64109
H = 64109

This question shows how nearly a great many trials
may be expected to agree with the probable mean: in
50,000 trials, it is nearly two to one against the
number of white balls differing from 30,000 by more
than a hundred.

In 100,000 tosses, between what limits is it 99 to 1
that the heads shall be contained ?

a@=1, b=1, n=50,000, p=99, g=1
100)99(:$9=H
t= 1°8215

447 °21
31

130) 13863°51(106°64
8x1lxIx 50,000 100)99(:99 = K

2

= 200,000

USE OF TABLES. 83

200,000 = 447°21 t=3'821
1°8215 sms
447-21 ihahen
814°6 407 *47

1 5
2)813°6 406°97 =z
406 8 =l

Answer. Between 50,000 — 407 and 50,000 + 407.

Now for the inverse method attached to the preceding
problem. If I. be totally unacquainted with the na-
ture of the events A and B, except only that one or
other, and not both, must happen every time, it is then
clear that, as the matter stands, it is to me 1 to 1 forA
against B, with avery great chance that, if I were better
informed, I should form a different opinion. At the same
time (page 10), I choose 1 to 1 as my rule of action,
because, though coming events may not justify my pre=
diction, I know of nothing to warrant my assuming
that the odds are in favour of A, rather than in favour of
B. A trial takes place, and A happens; it becomes im-
mediately most safe to assume that the odds for A against
B are 2 to 1, but still that safety is not very decided.
But if 1000 trials be made, and if A have happened
520 and B 480 times, I can then confidently say, that
the odds for A against B are very nearly, if not exactly
520 + 1 to 480+ 1, which is nearly 520 to 480. The
notion then formed has a strong presumption that it is
nearly correct.

Prosriem. In a+ 6 trials A has happened a times
and B 6 times: from which, if @ and 6 be considerable
numbers, it is safe to infer that it is a to 6 nearly for
A against B. What is the presumption that the odds
for A against B really lie between a—k to b + k and
at+ktob—k?

G2

84

Rute. (Taste I.)

Divide twice the product of
a and b by their sum, and ex-
tract the square root of the
quotient, by which divide &.
Then the last quotient being
t, the H of the table is the

ESSAY ON PROBABILITIES.

Rute. (Tasrz II.)

Having found the square
root, and divided & by it, as
opposite, from seven times the
quotient, take the hundredth
part of the quotient, and take
three tenths of the remainder.

Make the result t, and K in
the table is the probability re-
quired,

Suppose that in a thousand trials, A has happened
exactly 600 times, and B 400 times ; what is the pre-
sumption that the odds for A against B lie between 570
to 430 and 630 to 370? |

a=600, b= 400, k=

probability required.

2 x 600 x 400 = 480,000 1°369
”
480,000 _
woo 9°583
014
/ 480 =21°91 9569
ai} S6O ct :
21°91 28°707 t= 2: 871

Answer. About 95 to 5, or 19 to 1 in favour of the
odds being between the limits specified.

In the preceding problem, A and B have happened a
and 6 times; whence the most likely of all individual cases
is, that the odds for A against B area tod; or, in other
words, the result which has the strongest presumption
in its favour is, that

Probability of A was :
a+b.

Probability of B wrasse
a+b

Now we have found, in the preceding problem, the
presumption that

Probability of A lies between prea and —— ees
a+b a+b

USE OF TABLES; 85

or, which is the same thing, that

b+k
Probability of B lies between x
a

For since it is our hypothesis, that either A or B must
happen at every trial, whatever presumption there is that
the chance of A is x, there is the same presumption that
the chance of Bis l1—a&

But we might ask the following questions: A and B
having happened a and 6 times in a + 6 trials, what are
the values of the following presumptions P

k
1. That the probability of A lies between a4 and ey
a+b a+b
or its equivalent, that the probability of B lies between
b—hk b
octane
a+b a+b.
a k
2. That the probability of A lies between —— and —— oS
a+b a+b
. : b b—k
or that the probability of B lies between and —— -
a+b a+b

To solve these by the help of the following rule,
remember that, if @ be greater than b, it is more likely
that the chance of A falls short of a+(a+b) than ex-
ceeds it: and if @ be less than 0b, then it is more likely
that the chance of A exceeds a+(a+5) than falls short
of it.

Rute. First find the result of the preceding pro-
blem, and find from Table I. the H’ (p. 72) belong-
ing to the value of ¢t. Subtract this from the H”
derived from 0 in the table (which is 1°12844) ; mul-
tiply by the difference between b and a, and divide by
the product of the square root used in the preceding
problem and three times the whole number of trials:
call the result V. To one half of the result of the
preceding problem add V ; and from it subtract V: and
eall these results 3H + V and SH—V.

Then, if B have happened most times, }H + V is the

a 3

86 ESSAY ON PROBABILITIES.
a
presumption that the chance of A lies between ay and
a+

k
— or that the odds for A against B lie between ato b

and a+k to b—k. But in this case, 1LH—V is
the Prsanep Bion that the chance of A lies between
a—k

aoe and ae , or that the odds for A against B lie
between a—k to b+ and a to b.

But if A have happened most times, make H+ V
and 4H — V change places in the preceding paragraph,
every thing else remaining the same.

Exampie. In a thousand trials, A has happened
600 times, and B 400 times. What is the pre-
sumption, 1. that the odds for A against B lie
between 600 to 400 and 630 to 370; 2. that the
odds for A against B lie between 570 to 430 and 600
to 400?

From the preceding problem t=1°369, say=1°37 ;
the square root is 21°91, and H is -9471.

t=1°37 A 17036 1°12844 b—a=200
$A2 239 17268 3
3000 x 21°91 = 65730
H’ = +17268 *95576
200
65780)191°152(-0029 = V
4736 =4$H
4765 =$H+V
4707 =4H-—V

Hence (since A has happened most times), it is *4707
to °5293, or about 47 to 53, that the odds for A against
B lie between 600 to 400 and 630 to 370. And it is
"4765 to °5235, or about 48 to 52, that the odds for A
against B lie between 570 to 430 and 600 to 400.

The problems which the preceding part of this
chapter has enabled us to solve, are the determination
of the chance which exists (under known circumstances}
for the happening of an event a number of times which

USE OF TABLES. 87

lies between certain limits, and its converse. The
latter problem contains a consideration of some diffi-
culty, namely, the probability of a probability, or, as we
have called it, the presumption of a probability. To
make this idea more clear, remember that any state of
probability may be immediately made the expression of
the result of a set of circumstances, which being in-
troduced into the question, the difficulty disappears.
Thus, suppose a large number of urns containing
various proportions of black and white balls. Let
there be 100 urns, and let one of them only contain
equal numbers of black and white balls. If then
I lay my hand upon one of these urns with the intention
of drawing, it is, before the drawing, 99 to 1 against
my having placed my hand upon an urn from which,
in the long run, equal numbers of both sorts of balls
will be produced : the presumption black and
white balls have an even chance is only -},,; the pre-
ene that the probability of a white ball is 4, is
sy speaking of compound probabilities, writers have
employed six words synonymously—probability, chance,
presumption, possibility, facility, and expectation. Re-
jecting only the word possibility, as indicating a thing
of which there cannot be different degrees, the five
remaining terms have their advantages, each one
pointing out a peculiar and useful view of the main
idea. Thus the word presumption refers distinctly
to an act of the mind, or a state of the mind, while in
the word probability we feel disposed rather to think of
the external arrangements on the knowledge of which
the strength of our presumption ought to depend, than
of the presumption itself. When, therefore, having
observed an event, we want to know how strongly we
are to suppose that the observed event was preceded -by
a given arrangement of circumstances, the term pre-
sumption of probability is very appropriate. The word
facility applies particularly to the notion which we
form when we see one event happen more often than
a 4

88 ESSAY ON PROBABILITIES.

another, namely, that it is easier to produce the first
than the second. In our problems, however, the
facility is not that arising from art, but from previous
(it may be accidental) distribution of means. The
word expectation will be applied throughout this work
to that state of things for the production of which
there is an even chance. If (p.79), 6000 throws be
made with a die, it is an even chance that the number
of aces lies between 981 and 1019: the odds are
against any smaller amount of departure on both sides
of the probable mean, and against any greater amount ;
this is then our expectation of the number of aces.

When one of two possible events happens oftener than
the other, it being understood that one, and only one,
can happen each time, we are led to suppose that the
excess of one event is the consequence of some arrange.
ment which would, had we known it, have made us
count that event more probable than the other. If A
or B must happen, and if in a thousand trials the As
outnumber the Bs very much, we feel perfectly cer-
tain that such must have been the case. The theory
of probabilities confirms this impression, as will appear
by the solution of the following

Propitem. In a+b trials, the number of As was
a, and that of Bs was 6. If a exceed b considerably *,
required the presumption that there was at the outset
a greater probability of drawing A than of drawing B,
in any one single trial P ;

Ruue. Divide the difference of a and 6b by the
square root of twice their sum, and let the result be t..
Find (page 72) the H! corresponding to t: multiply
the result by the sum of a and b, and divide by the
product of 8, ¢, and the square root of the product of
aandb. The result subtracted from unity gives the
answer required. Suppose, for instance, that out of
50 trials A occurs 32 times, and B18 times. Then,

* In order that the result may be very correct, a must exceed 6 so much

. that the excess of a above b, multiplied by itself, may considerably exceed
the sum of a and 4,

USE OF TABLES. 89

50x2=100, Y100=10 32-18
10
H’=:15891 15891 x 50=7:9455
V 32 x 18=24, 8x 24% 1°40=268°8
7°9 divided by 268°8 is 58; nearly: 1— 8, is 384

=140=

Hence it is about 261 to 8 that A was more probable
than B.
AppiTionau Rute. When a and? arenearly equal,
find t, as in the last rule; find H (not H’) corresponds
ing to t, add 1, and divide by 2: the result is the pro~
bability required. ,

The additional rule belongs to the more important
case of the two, namely, that in which A has not hap-
pened so much oftener than B as to justify an imme-
diate conclusion that it was the more probable event of
the two. Suppose, for instance, that A has occurred
10,100 times out of 20,000 trials, and B 9,900 times:
then t = 200 divided by 200, or 1; to which H is
*843, and this increased by 1, and the result divided by
2, gives ‘922. It is, therefore, about 114 to 1 that A
was the more probable.

The preceding solution can be applied to various
species of observations ; of which we shall see more
hereafter. The following may be considered as closely
connected with it. If we make two different sets of
trials, in circumstances which we suppose to be the
same, it will generally happen that the As will not bear
the same proportion to the Bs in both sets. If, for
instance, we find 1000 As arrive in 2000 trials, the
odds are very much against the arrival of exactly 5000
As in anew set of 10,000 trials, though the expectation
is that something near that number of As will arrive.
Suppose'that the first and second sets of trials give—I1st,
50 As, 30 Bs; 2nd, 112 As, 61 Bs.

In the second set the As bear a larger proportion to
the whole than in the first: and our present question is
what presumption thence arises that there is some dif-
ference of circumstances between the two sets, which

e10) ESSAY ON PROBABILITIES.

gives A a greater facility in the second than in the first,
or a greater probability of being drawn at any one trial ?
Or, if in a first set of a + 6 trials, A happen a times
and B happen 6 times; andif in a second set of a’ + b!
trials, A happen a times, and B happen Db’ times ; and if
a' be a larger proportion of a’ + 6’ than a is of a + b;
required the presumption that there was a greater chance
of drawing A at a single trial in the second set than in
the first P

Rue. Divide the cube of the sum of a and b by
twice their product: do the same with a’ and 6’: mul-
tiply the two results together, and add them together :
divide the product by the sum, and extract the square
root of the quotient.

Divide a’ by a’ + 0’, anda by a + b, and subtract
the less result from the greater. Multiply the difference
by the square root previously found, and let the product
be @ Then the H corresponding to ¢, increased by 1,
and divided by 2, is the presumption required.

In the example a = 50, b = 30, a’ = 112, b’ = 61.

80 x 80 x 80= 512000 2x 50x 30=3000, 212000 ~ y794
3000

173 x 173X173 = 5177717 2x 112 x 61 = 13664 an = 3789
170°7 x 878°9 = 64678'23, 170°7 + 378°9 = 549°6
64678'23 = 117°7, 4/1177 = 10°85
5496
112 — 50 — -6474\— 6250 = 0294
"0224 x 10°85 = *24904 = t, H = “2689

4(H + 1) = ‘635, the probability required ; and it is
therefore about 16 to 9 in favour of the excess of As
at the second set of trials not being accidental fluctua-
tion, but arising from some new circumstance or dif-
‘ferent arrangement of the old ones.

If, in 1000 trials, A should happen 520 times, and
B 480 times, there is strong presumption that in any fu-
ture number of trials the whole number will be divided
among As and Bs nearly in the proportion of 520 to

USE OF TABLES, G1

480. But this is not the same set of circumstances as
that of the problem in page 77. We are there sup-
posed to know exactly in what proportion As and Bs are
contained in an urn; and with this positive knowledge
we can ascertain the probability of drawing any given
number of As in a given number of trials, In the
present instance we do not know the contents of the
urn, but only the result of a certain number of drawings,
from which we can draw presumptions, as in page 53.
about the whole contents. The determination of chances
relative to a new set of trials depends upon two risks in
the latter case, and upon one only in the former. The
latter problem is therefore more complicated in its prin-
ciples though not so in its results.

Let us suppose two different persons, John and
Thomas, thus situated with respect to the contents of an
urn; John knows that there are as many As as Bs;
Thomas has observed a hundred successive drawings, of
which (so let it have happened) fifty have given A, and
as many have given B. That which John knows is
rendered not improbable to Thomas by the result of
the trials, while the same result would have been thought
not unlikely beforehand by John. But there is this
difference between their degrees of knowledge, that John
has the certainty of a fact (the equality of As and Bs),
of which Thomas can only say that the fact, or some-
thing near it, is extremely probable. No one could
argue with John against any particular venture in such
a lottery upon the ground of the possibility of the As
much exceeding the Bs ; while with Thomas it might
be urged as possible, though not probable, that the
former might exceed the iatter a hundred-fold. Again,
suppose John and Thomas, having equal fortunes, are
disposed to venture as far as produce woulda warrant,
upon the results of a hundred (to Thomas a second
hundred) trials. It is obvious to common sense that
Thomas must not venture so much as John; for he
runs a larger risk, seeing that he assumes as an average
result what possibly may have been a rare occurrence.

92 ESSAY ON PROBABILITIES.

The following is the rule pointed out by the theory of
probabilities: —The expectation of fluctuation should
be greater to a person who proposes to try g new in-
stances, upon the assumption that p preceding instances
have fairly represented the Jong run, than it should be
to another person, who knows in what proportions the
As and Bs really exist ; and greater in the proportion of
the square root of p augmented by gq to the square root
of p. Thus, if in the preceding case, John and Thomas
propose to embark in a matter which depends on 300
more trials,the proportion of the squareroot of 100 + 300
to that of 100, being that of 2 to 1, it follows that,
whatever reason John may have to guard against the
possibility of 300 drawings giving w more than 150 As,
Thomas has as much reason to guard against 2~ more
than the same number.

Prospiem (to be compared with that in page 77.).
When a + 6 trials have happened to give a As and 6
Bs, required the chance that in m times a + b new
throws, the number of As shall not differ from na by
more than J.

Ruue. Divide one more than twice 7 by a square
root to be immediately mentioned, and the quotient being
made ¢, the value of H in Table I. is the probability re-
quired.

The square root in the; The square root in the
former problem was that of the| present problem, is that of

product of 8, n, a, and b|the product of 8,2, n+1, a,
divided by a+. and 6 divided by a+ 6.

The additional rule in page 81. may also be applied
verbatim, the square root now meaning the second
square root above given ; and the inverse rule (p. 82)
may be applied in exactly the same way.

Exampxe. In 600 drawings A occurred 100 times,
and B 500 times ; what presumption thence arises that
in 6000 more drawings A would occur somewhere be-
tween 1000 — 50, and 1000 + 50, or 950 and 1050
inclusive? (See page 77 for the corresponding pro.
blem.)

ON THE RISKS OF LOSS OR GAIN. 93
#=10, a=100, b=500; n (n+1) ab+(a+b)=73333
Pras =n °S375 at
270°8
H = °4 very nearly, or about two to three against the
proposed event.

Having thus shown the use of the tables at the end
of this work, in the solution of complicated questions, I
now proceed to the application of the theory to questions
involving loss and gain.

/73333=270°8; 21+1=101;

CHAPTER V.
ON THE RISKS OF LOSS OR GAIN,

Tue proverb which advises us to throw a sprat to catch
a whale, shows that mankind consider a chance of a
gain to be a benefit for which it is worth while to give
up a proportionate certainty. The principle on which
depends the determination of the amount which it is
safe to hazard, must vary with the circumstances of the
person who runs the risk. A man should not hazard
his all on any terms ; but in ventures the loss of one of
which would not be felt, we may suppose the venturer
able to make a large number of the same kind; in
which case, the common notions of mankind, reinforced
by the results of theory, tell us that the sum risked
must be only such a proportion of the possible gain
as the mathematical probability of gaining it is of
unity, For instance: suppose I am to receive a shilling
if a die, yet to be thrown, give an ace; in the long
run, an ace will occur one time out of six, or I shall
lose five times for every time which I gain. I must
therefore make one gain compensate the outlay of six
ventures, or one sixth of a shilling is what I may give

G+ ESSAY ON PROBABILITIES.

for the prospect, one time with another. But } is the
probability of throwing the ace.

Principte. Multiply the sum to be gained by the
fraction which expresses the chance of gaining it, and
the result is the greatest sum which should be given for
the chance.

Propiem. Iam to gain by the throw of a die as
many pounds as there are spots on the face which is
thrown, with this exception, that I am to lose as many
pounds as there are spots on both the face in question
and on that of another die which is thrown at the same
time, if the two give doublets. What should I give
for the chance in question ?

Call these dice P and Q: then the chance of an ace
from P, and something not an ace from Q, is the pro-
duct of 4 and 2, or <5; and the same for any given
throw from P, and not the same from @. Consequently,
the chance of winning, independently of the chance of
losing, is worth

Bx l+ixQ+ Rex St Hx atx 5+ 3 x 6,

or 3; X 21, or £214. But if the chance of winning
must be paid for, the chance of losing must be paid.
Now the chance of throwing any one pair of doublets
is the product of 1 and 4, or ,4.: whence the sum I
must receive if I pay £244 (or not pay out of the £211,
which is the same thing) is

Bxe2+ex4etd x Gtx Std x 104+ 34 x 12,

or £1}. Consequently, I must pay only £13. If I were
to stand a million of such hazards, paying £15 for each,
I might expect my ultimate gain or loss to be extremely
small, compared with the whole sum risked. If I had
besides a very small profit upon each, I might be sure
of winning.

We have in the last chapter considered the amount of
fluctuation for which there is a given probability: we
will now look at the percentage of fluctuation, that is,

ON THE RISKS OF LOSS OR GAIN 95

at the proportion which the fluctuation bears to the whole.
It is for want of consideration on this point that many
notions prevail which are utterly at variance both with
the daily evidence of facts, and the results of exact in-
vestigation.

Name any sum of money as a probable gain or loss
of a commercial transaction, and we imniediately pro-
ceed to compare it with the whole capital by which it is
to be borne, if lost. A hundred pounds is nothing in
one case, because it is only a shilling per cent. on the
outlay ; in another, the same sum is important, because
it is ten pounds per cent. The importance of a gain
or loss, then, depends upon the relative, and not on the
absolute, value of the sum in question. Similarly, ina
Jarge number of transactions of the same sort, the
number of them which may go against previous calcu-
lation is only important as compared with the whole
number in question. The following is the correct prin-
ciple, namely, that the percentage of fluctuation for
which there is a given chance, varies inversely as the
square root of the whole number of trials. Suppose,
for instance, I find that on a certain hundred risks it
is an even chance that one out of twenty goes against
calculations made by the preceding methods; then if I
take four times as many trials, it is an even chance that
one out of forty only will disappoint the calculations ;
if nine times as many, one out of sixty, and so on.
Thus it appears that the chances may be made to be
against any named percentage of fluctuation, however
small, by sufficiently multiplying the number of risks.
The probable amount of fluctuation increases, but not
so fast as the number of risks, in such manner that the
proportion of the probable fluctuation to the whole
diminishes.

Gambling, according to the common notion of the
word, means the habit of risking considerable sums in
hazardous transactions. The gambler, properly so called,
at cards and dice, and the jobber at the Royal Exchange,
are called by the same name. Moral considerations

36 ESSAY ON PROBABILITIES.

must induce us to regard with an evil eye the person,
whatever be the name of his occupation, who thrives
by the actual losses of others, in transactions which are
not commercially beneficial to the community. I am not
prepared to say that stock-jobbers deserve to be included
in this class ; it may be that the acquisition or sale of
bond fide investments may by their means be rendered
more easy to be obtained, and that they themselves may
be a useful circulating medium between those who
really wish to buy and sell for their own purposes.
However this may be, the character of a gambler
has no claim, when he is skilful, to the sort of
respect which we pay to those who risk, The man of
cards and dice, if he be cool and stick to his principles,
secures a certainty to himself, and throws all the hazard
upon his opponents: taking care never to risk too large
a proportion of his means, and thus always enabling
himself to take the benefit of the long run, he manages
to play upon terms slightly unequal, and, of course, in
his own favour. He runs a greater risk in games of
mixed skill and chance than in games of pure chance
alone. The latter he does not play at, unless he know
the chances to be in his own favour ; while in the for-
mer it is next to impossible that he should always be
more than a match for every opponent. The keeper of a
gambling house has the surest game imaginable: the play
is so managed that there shall be some chances more for
the bank, as it is called, than for those who play against
it; care is taken to provide a sum sufficient to stand a
considerable succession of losses; and the preceding
principles will enable us to show that no individual re-
sources can stand against a large fund thus used. The
following problems will illustrate this.

Prosiem. An indefinite number of successive ha-
zards are tried of the following kind :—One of the
events, A, B, C, &c. must happen at every trial, and
each event brings with it a specified gain or loss,
What may be expected to be the result of continuing to
play a yery great number of trials ; or what, in the long

ON THE RISKS OF LOSS OR GAIN. O7

run, may be expected to be the average produce or loss
per game P

Rute. Multiply each gain or loss by the probability
of the event on which it depends; compare the total
result of the gains with that of the losses: the balance
is the average required, and is known by the name of
the mathematical expectation. When the balance is
nothing, then the play is equal.

For example, a person plays against a bank in the fol-
_ lowing manner :— Equal stakes of one shilling each are
laid down, and if head be thrown twice successively he
wins, or if tails be thrown twice he loses. But if
head and tail be thrown, then another throw is to be
made, by which, if it be head, the player only recovers
his stake, but wins nothing: while, if it be tail, he
loses his stake. In such problems, the stake is a mere
evidence of solvency, with which the mathematical
question has nothing to do. Let us suppose, then, that
both parties keep their money until it is called for by the
result. The events on which the player gains or loses
are as follows (H stands for head, T for.tail) :—

HH gains him a shilling.
TT, HTT, THT lose him a shilling.
HTH and THH give no result.

The chance of HH is 4, and the same of TT; while
for each of the st HTT, THT, HTH, and THH,
the chance is 1. Hence the value of the prospect of
gain is ; of one shilling ; that of loss 1 + 4 + 4, or § of
a shilling : consequently the balance against the player
is + of ashilling each time, and this he would certainly
lose in the long run; that is to say, the number of times
he loses would eventually so far exceed the number of
times he wins as to make the play cost him three pence
per game.

Every risk of loss must then be compensated by an
equal chance of gain, when the play is equal. But it
does not follow that equal play means prudent play ;

H

98 ESSAY ON PROBABILITIES.

and for the following reason. Prudence requires that
no one should expose himself to great risks of loss, and
does not accept it as an excuse that there was a remote
chance of enormous gain, or that another had the same
chance of the same gain or loss. Again, the mathe-
matical expectation is derived from the result which, as
ean be shown, will be produced in the long run: con-
sequently no one can prudently play, even with the play
in his favour, unless he continue the occupation through
such a number of trials that he may reasonably expect
an average of all sorts of fortune. And though I have
hitherto appeared to speak only of games of chance, yet
precisely the same considerations apply to mercantile
speculations, and to every species of affair in which no
absolute certainty exists. If any possible event enter
into the play which, from the nature of the game, cannot
often occur, and if a stake be made upon the arrival of
that event, proportionate to some enormous benefit which
it is agreed that event shall secure, then prudence re-
quires that the game shall be very often repeated, or, if
that cannot be done, that it shall not be played at all.
There is a celebrated case known by the name of the
Petersburgh problem, which is one of the most in-
structive lessons in this subject, both on account of its
paradoxical appearance, and also because very eminent
writers have considered it as a sort of stumbling-block,
and have endeavoured to evade the conclusion. Con.
dorcet and others have taken a proper view of the subject;
while among those who have considered the problem as
an anomaly, we may instance D’Alembert.

If p, q, and 7 be three fractions whose sum is unity,
it follows that we may suppose three events, one of which
must happen, and neither of the others, and of which the
chances are p, g, and 7: and similarly of more fractions
than three, whose sum is unity; or of any number of
fractions, however great, provided their sum be unity.
Let us, then, take an infinite number of fractions whose
sum is unity: either of the following series will do.

ON THE RISKS OF LOSS OR GAIN. 99

1 1 1 9 9 9

£4 8 te ty &O ¥% 80 TOO Toss &C-
K) 4 5 1 3

4 2 ys 3s ee & $ te wm &e.

Let the game be as follows : —- The events which may
happen at every trial are E,, of which the chance is 3 ;
E,; of which the chance is 4+; E,, of which the chance
is 4; and so on, ad infinitum. And one of these must
occur. The bank engages to give 2/. if E should turn
up, 4/. for E,, 8/. for E,, 160. for E,, and so on,
ad infinitum.— What should the player give to the bank
for one trial? Write the several possible gains in a row,
and underneath each the chance of its being won, as
follows :—

£2 4 8 16 32 64 128, &c.
3° 3° & (fb = te Se.

Multiply each gain by the chance of gaining it, and
each result is 1 ; consequently the mathematical expect-
ation of the player is unity repeated an infinite number
of times, or an infinite amount. No sum, then, how-
ever great, can compensate the bank for its risk. The
Petersburgh problem realises the preceding supposition as
follows: —A halfpenny is tossed up until a head arrives,
which is the event in question. If this happen at the
first toss, the player receives 2/.; if not till the second,
4j.; if not till the third, 8/., and so on. Now, H
standing for head and for tail, the chance of H is 3;
of TH, 4; of TTH, 3; of TT TH, 54,3 andso on,
But can it be believed, that if I am only to throw until
head arrives, and to receive 2/., or 4/., or 81., &c. ac-
cording as this happens at the first, second, third, &c.
throw — can it be believed, you will say, that this pros-
pect is even worth 100/.; and is it not altogether mon-
strous to say that an infinite amount of money ought
to be given for it?

Firstly, I will advert to a large number of trials
which was actually made. Buffon tried 2048 experi-
ments, or sets of tosses, the results of which were as
follows:—In 1061, H appeared at the first toss; in

H 2

100 ESSAY ON PROBABILITIES.

494, at the second; in 232, at the third; in 137, at
the fourth; in 56, at the fifth; in 29, at the sixth; in
25, at the seventh; in 8, at the eighth; and in 6, at
the ninth. Let us, then, compute the amount which he
would have received if he had bond fide played all these
games on the preceding terms, _

1061 x 2 = 2122

494 x 4= 1976

232 x 8 = 1856

1s7 x “16 = 2192 The 2048 games would have
66x 82 = F792 given 20,114/. or nearly |
29 x 64 = 1856 107. per game, one game
25 .x -128 =. 3200 with another.
8 x 256 = 2048
G6 x S12 = SO7Z

No person would stake at this game for a single trial,
upon the prospect of head being deferred till the ninth
throw. Nevertheless, in this instance, it appeared that
out of 2048 trials, such a rare occurrence happened often
enough to realise more than any other, with one excep-
tion. If Buffon had tried a thousand times as many
games, the results would not only have given more, but
more per game. A larger net would have caught, not
only more fish, but more varieties of fish ; and in two
millions of sets, we might have expected to have seen
_ cases in which head did not appear till the twentieth
throw. Let us turn back to page 43, and inquire, by
the rule there given, in how many trials it is 10,000 to
1 that head will be deferred till the twentieth throw.
Out of 22°, or 1,048,576 cases, representing the number
of different arrangements which may happen in 20
throws, the arrangement in question is but one; it is
then 1,048,575 to 1 against its arrival in any one given
trial. Look in the Table opposite to “ 10,000 to 1,”
and we find 921: multiply 1,048,575 by 921, and
divide by 100, which gives 9,057,375. It is then
more than 10,000 to 1 that head is deferred till the
twentieth throw somewhere in ten millions of trials, and
more than an even chance that it is found to occur in
seventy thousand trials. Thus the reader may readily

ON THE RISKS OF LOSS. 02 GAIN _-- 101

conceive that with unlimited license of ‘proceeding in
this play, the player might continue until he had realised
not only any given sum, but any given sum per game:
a result which is indicated by the application of our rule,
when it tells us that the mathematical expectation of
the player upon a single game is infinite.

The result of all which precedes shows us that great
risks should not be run, unless for sums so small that
the venturer can afford to repeat them often enough
to secure an average. But it should seem asif we were
thus told either not to gamble at all, or else to play in-
cessantly. With a little reservation, this is true; the
stake must be lowered, and more games played, instead
of risking a large fraction of the whole upon one game.
It is better to buy the sixteenth of sixteen different
tickets than to stake all upon one ticket ; and this even
though it should be better than either not to buy at all.
It is more prudent to play twenty games, staking one
shilling upon each, than to stake a sovereign upon one
game. Lay a proper proportion of the whole capital
upon any hazard, and stipulate for as many trials as you
please, and it will follow that with any mathema-
tical advantage, however trifling, in your favour, you
must come off a winner. The mistake committed by
those who attempt to gamble with professional men, is
twofold: firstly, they set out upon unequal terms; se-
condly, if the terms were equal, their stakes would be
too large a proportion of their means. That the terms
are unequal may readily be supposed, and will presently
appear. No bank or individual gamester can play on
fair terms, without losing as much as he wins in the
long run. But even in such a case, the player of
superior fortune has a great advantage over his anta-
gonist, unless the stake be very small. If A with
twenty guineas engage B with forty, all other things
being equal, and if they are to play on until one or
other has lost all, it is obviously much more likely that
A shall lose his money before B, than the. converse.
If the play be unequally in B’s favour, as well as the

H 3
.

102 So 9) (oo BSSAR ON PROBABILITIES.

largeriess of the’ fund, “then’it is still more against A in
any given succession of games. The truth is, that to a
young man who is determined to gamble, whether at
one of the private receptacles in London, or the (till
lately) recognised saloons of Paris, it is of little con-
sequence whether his stakes be high or low, except in
this particular, that a longer process of ruination will
give him more chances of seeing his error. The play is
against him in both cases, and sooner or later he must
be ruined. Nor if his means be ever so great, could he
make use of them, against the banks in Paris, at least.
Those who conducted the play at the Palais Royal
were perfectly aware of the necessity of not staking too
much, and limited not only the amount of each stake,
but also the number of persons whom they would en-
gage at once. The consequence was, that though they
played with perfect fairness (inasmuch as the inequality
which existed in their favour was known to, and recog-
nised by, their opponents), they gained large returns
upon their capital, besides paying a considerable duty
to the government.

A gambler (meaning a bold venturer, which the term
commonly implies) ceases to be such when he makes
his stakes bear a proper proportion to his capital, and
takes no hazards which are unduly against him. If,
then, a government should attempt to discourage the
acquisition of great losses and gains, by limiting the
number of hazards which an individual should be allowed
to take, it might defeat its own object; and this is the
case with our law, as it stands at present. In order to
prevent individuals from gambling in life-insurance, the
legislature has declared that A shall not insure the life
of B, unless he have what is called an insurable interest
in that life ; that is, unless A have some pecuniary in-
terest in B’s continuing to live. The insurance offices,
for the most part, have virtually, and very wisely,
refused to live under this law, by paying all fair claims
without questions asked. But supposing that the law
were enforced, its effect would be as follows, It is

a

ON THE RISKS OF LOSS OR GAIN. 103

tolerably easy to create a bond fide insurable interest on
a few lives, while it would be difficult and attended with
danger of detection to do the same with many lives.
Under the system, then, proposed by the law, it would
be easy to gamble, but not easy to carry speculation to the
extent which would make it cease to be gambling. Allow
the venturer to extend his traffic, and he will soon begin
to feel the average, not to his gain but to hisloss. For
the mathematical advantage is in favour of the insurance
offices, which are sure to gain in the long run. If, then,
the law had been intended to save the gambler from
certain loss, and to make it a real toss up, it would have
been rational, considered as means; but if, as I ima-
gine, it was meant to hinder immoral gain, a more futile
contrivance can hardly be conceived.

It must be remembered that, in the long run, events
will happen in proportion to the chances of their happen-
ing in a single trial. We see this result in Buffon’s
trial of the Petersburgh problem, for which I write
down the numbers of cases as they did arise, and un-
derneath as they would have arisen, one time with an-
other, if a great many series of 2048 trials each had
been made.

2048 1061 494. 232 187 56 29; 25 8 6
2048: 1024 S12 266..1298. 64.38.. 36 8 ..%

This rule, however, will only apply when so many
cases are taken as will produce a great many of every
event to which reference is made. For instance, in
2048 trials, one time with another, we can only expect
the deferment of head till the seventh throw 16 times ;
the result gave 25 times, or 50 per cent. more than the
probable average. But the occurrence of head at the
first throw, which, one set with another, would have
occurred 1024 times, did really occur 1061 times, or
3 per cent. too many times. If we remember that in
the long run, and on 2048 trials, we might expect two

H 4

104 ESSAY ON PROBABILITIES.

sets in which head should not appear till the tenth throw, —
and one in which no such thing should take place till
the eleventh, and if we calculate the total amount
which would have been realised had the average case
occurred, we shall find it to be £11 per game. In the
experiment in question, it would have produced £10. In
precisely the same way, sets each consisting of 2” games
would have realised £n per game (2048 is the eleventh
power of 2).

-I now come tothe estimation of the chances of
fluctuation in loss or gain, meaning by fluctuation
any departure from that general average to which the
results of more and more trials will continually approach.
It has been assumed in what precedes, that the propor-
tions which the fluctuation will bear to the whole will
diminish without limit as the number of speculations
increase. The following problems are easy deductions
from those in the last chapter.

Prosiem. It is known to be a to bfor A against B.
A is an event which brings a loss or gain of g pounds ;
B is another event which brings a loss or gain of h
pounds. What is the general average of such trials ;
and what is the chance that in n times a + 3 trials, the
result as to loss or gain shall differ from the general
average by not more than v pounds.

Rute. Find g times a, and h times b, and if g and
h be both gains or both losses, take their sum ; but it
one be a gain and the other a loss, take the difference,
counting it gain or loss, according as the term which
contained the gain or the loss was the greater. Multiply
the result by n, which gives the most probable total result
(call this M). The general average is the n (a + 6)th
part of this; or, more simply, the (a + b)th part of
the balance of g times a and h times b. Take the differ-
ence of g and h, if of the same name, or their sum, it
of opposite names, and by it divide v. Take one more
than twice the quotient. Having found this result,
divide it by a square root immediately to be described,
and let the quotient be ¢.’: Then the value of H in

ON THE RISKS OF LOSS OR GAIN. 105

- Table I. is the probability that the resulting gain or loss
shall lie between M + v and M — v pounds.

If it be absolutely known that the chances are as a
to b for A against B, then, as in page 81, the square root
is that of the product of 8, n, a, and b, divided by a+b.
But if all that is known be that in a + b previous spe-
Jations, a gave A, and b gave B, then, as in page 92,
the square root is that of the product 8,n,n + 1, a,
and 6, divided by a + 0b.

Exampne. It has been observed, that of 100 specu-
lations, 70 yielded a profit of £20 each, and the re-
mainder a loss of £25 each. Whatis the probability
that in 150 more such speculations the total result shall
not differ by more than 100 from its most probable
amount P

a= 70, b=30, n= 11, 9 = £ 20gained, h = £ 25 lost,
» = £100.
g times d = 1400 gain

$50 —
A caes be S50 ae = £ 6} general average of gain.

650 oe x 11 = £975 probable total.

g.= £ 20 gain 2
hc ® 25 lous AEB on, 2°2299 : 2°29922 x 2+1= 5°4444
Add £46.
8xnxn+1lxaxb=8xI1}x2}x 70x 30=68,000
1-8 bs Sean Meier eee 5+4444
—_—_ = = 2 =: °217 =
100 630, a/ 630 5°100, ——— 25°] 7 t

Table ay if t = 217; H = 241.

Hence it is more than 3 to 1 against the result lying
within the given limits.

Prospiyem. All things remaining as in the last pro-
blem, what is the amount of departure from the pro-
bable total for being within which there is the given
odds p to q ?

Rutz. Turn p divided by p+ q into a decimal
fraction, and find it in the column H of Table L.,
taking out the corresponding value of t. Multiply t by

106 ESSAY ON PROBABILITIES.

the square root above mentioned, and having subtracted
1, divide the remainder by 2. Multiply the quotient
by the difference or sum of g and h, according as they
are of the same or different names, and the product is
the answer required.

Exampie. In the preceding example, within what
departure from £975 is it 10 to 1 that the result shall
be contained ? |

p= 10,q=1,p + (p + g) ="9091, ‘9091 x 25'1 = 22°8184.
1 (228184 — 1) = 10:9092 10°9092 x 45 = 490-91.

It is, then, 10 to 1 that the balance of 150 speculations
shall lie between 975 + 491 and 975 — 491 pounds, or
14667. and 4847. Even such a case shows the effect of
multitude in diminishing risks. The possible extremes
of the problem (or the result of the problem itself, if
we supposed one speculation instead of 150) are a gain
of 3000/., and a loss of 37501.

I will now add an example which will tend to show
the ultimate effect of gambling against a bank with a
slight mathematical advantage in its favour. Suppose
the game is such, that at each trial it is 30 to 29* that
the bank shall win, the stake on both sides being one .
sovereign. Here a + 6b is 59, and making n a whole
number for convenience of calculation, let m = 50, or
let 2950 games be tried. ‘The bank has each time a
mathematical advantage (page 97) of 29 — 28,or 35
of a sovereign, and will, in the long run, realise £ 50
upon 2950 games. What are the chances in favour of
the individual fluctuation of this one set of 2950 games
leaving the bank without any profit, and with more or
less loss? To apply the preceding rule, we must first
ask what are the chances that the departure from the
probable total of 507. shall not exceed 50/.; that is,
that the bank shall realise between O/. and 1002 Here
we have

* This supposition is more in favour of the player than is often the
case.

ON THE RISKS OF LOSS OR GAIN, 107

a= 30,b6=29,2 = oka £] — h=Z£l lost v=£ 50,
g times a = 30 gain
A times 6 = 29 loss ls =2£55 = general average of gain,

See

1 gain x 50 = £ 50 probable total.

a 50
agg pag 5 = 25; 2542+ 1 = 51.

Add £ 2.

In this case the probability is absolutely given. The
square root is therefore the first one mentioned.

&8xnxaxb=8 x 50 x 30 x 29 = 348,000

348000
39

ee 51
= 5898'3, “5898-3 = 76°800, can = Cet mt.
Table I., if t = -664, H = °652.

Consequently 1000 is the chance that the bank shall not
lose, but shall gain something less than £100; and
consequently the chance that the bank shall either lose,
or gain more than £100 is -348. On account of the
nearness of 30 and 29, the results last mentioned are
nearly equally probable, and it is near enough for our
present purpose, to say that =4,/;4, is the chance of the
bank gaining more than £100, the supposition being
against us; for it is more likely that the bank should
gain more than £100 than that it shouldlose. Hence
it follows, that the chance of the bank gaining on 2950
games is more than -8*°,, or about five to one. If such
be the case with a bank much less unfairly constituted
than is often the case, against a player who can not only
command £ 2450, but who has the prudence to deter-
mine that he will only play 2950 games, at a stake of
£1 for each game, what must be the chances against
those who risk a larger proportion of their means at
more unequal play, with a determination to win —that is,
to go on till they are ruined ?

The inequality of means is an important consider-
ation in calculating the chances of two antagonist game-
sters. If two persons, with equal means and equal
chances, play for equal stakes, it is an even chance

108 ESSAY ON PROBABILITIES.

whether A shall ruin B, or B shall ruin A; but that
one or other will ultimately be ruined is certain. Sup-
pose each party to have a hundred guineas, the stake
being one guinea, and suppose two millions of games
are to be played. ‘The most probable individual case is
that each shall win a million of games; but if the
fluctuation amount to 100 in favour of either, the other
is ruined. Now, page 81, the probability that the
number of games won by A shall lie between a million
+100 and a million — 100 is ‘112, which is therefore the
chance that neither player shall be ruined. Consequently,
it is about nine to one that one player or other is ruined
or more than ruined in two millions of games. And
the chance is even greater than this: for the preceding
method of treating the problem supposes the players
not to balance their account till two million of games
have been actually played, so that one piayer or the
other may have been repeatedly playing on credit. The
same rule may be easily applied to any inequality of play,
the fortunes of the players being equal ; and the result
is, 1. that ultimate ruin to one or other player is certain ;
2. that, if the stake be a sufficiently small fraction of
the player’s income, the number of games which must
be played to render probable the ruin of either may be
made as large as we please. There are but two con-
ditions under which gambling can be prudently followed
as an amusement — small stakes and equal play. In
games of pure chance it is possible to obtain the latter,
and almost impossible in games of mixed skill and
chance. Unfortunately, the stimulus of gambling, a
combination of suspense and hope of large gain, cannot
be obtained upon any terms which prudence would
sanction.

When two players, of unequal fortunes, play together
for the same stake, however equal the play may be, the
larger fortune has an unfair advantage. To estimate
the amount of the disadvantage, proceed as follows,

Propuem. ‘Two players, A and B, having funds o
m and n times their stake, play a game, at which it isa
to b that A wins, or } to a that B wins. — What is the

ON THE RISKS OF LOSS OR GAIN. 109

chance that in the long run B will ruin A, and that A
will ruin BP

Rute l. If a and bd be equal, it is m ton that A
will ruin B, and 7 to m that B will ruin A.

Rute Il. If a and 6 be unequal, let M represent
the difference of the mth powers of a and of 6, multiplied
by the nth power of a, and let N represent the difference
of the nth powers of a and of 6, multiplied by the mth
power of 6. Then it is M to N that A ruins B, or
N to M that B ruins A.

N.B.—If m and vn be considerable, it is almost im-
practicable to apply the above rule without the aid of
logarithms. When 0b is many times a, take the nth
power of a for M, and the nth power of b for N.

Rute III. When m and n are equal, it is as the
mth power of 6 to the mth power of a that B wih
ruin A.

Rute [V. If the means of both players be unlimited,
then it is certain the player who has odds in his favour
on a single game will, in time, gain any sum, however
great ; if the means of the stronger player be unlimited,
then it is certain he must at last ruin the weaker,

Rute V. If the means of the weaker player be
unlimited, and those of the stronger limited, the chance
that the latter will in time win any sum, however great,
from the former, is as follows. Let B be the stronger
player (that is, b greater than a), and let him begin
with 7 times his stake, while A has unlimited means:
then it is the excess of the nth power of 6b over that of
a to the nth power of a that B gains any sum, in the
long run, from A, and vice versd for A ruining B.

Exampie. A has 5/. and B 3/., and they play ata
game for which it is three to two that B shall win any
one game.— What are the chances for the ultimate
success of each player?

a=2,b=3,m=5,n=3
M or (35 — 25) x 23is (243 — 32) x 8 or 1688
N or (33 — 23) x 35is (27 — 8) x 243 or 4617:
It is therefore 4617 to 1688 that B shall ruin A.
A gaming bank must be considered as a player of

110 ESSAY ON PROBABILITIES.

limited means, playing against all who choose to enter,
that is, playing against unlimited means. It is,
therefore, essential to its existence that some mathe-
matical advantage should be allowed, even more than
is necessary to reproduce the expenses of its ma-
nagement. What I have hitherto said on the subject
refers to the relation between the bank and the indi-
vidual player against it. but considering the former as
the antagonist of all who choose to play, it absolutely
requires the protection of a mathematical advantage.
But having this advantage, it must, in the long run,
ruin its individual opponents ; so that bankruptcy to
itself, or degradation and suicide to its customers, are
the initial conditions of its existence. But since the
banks flourish, it is plain that whatever advantage is
necessary to their continuance, is really obtained by
them ; and I shall now inquire how much this advantage
must be in several cases.

Examine (Rule V.) It is 30 to 29 for the bank
upon each game, and the bank stakes the tenth part of
its means at every game.— What are the chances of its
perpetual continuance? (6 = 30, a = 29, n = 10).

30!° = 590,490,000,000,000
2910 = 420,707,233,300,201

169,782,766,699,799.

Answer. About 170 to 421, or such a bank would
not be likely to last; that is, in the long run, only 170
out of 421 such banks wouid avoid ruin..

Proprem. What is the mathematical advantage
which a bank must have, in order that its permanent
continuance may have k to 1 in its favour ; the sup-
position being that the bank stakes the nth part of its
means at every game?

Ruue. The odds in favour of the bank, on a single
game, must be the nth root of l +k tol. Thus if,
judging by the experience of the Parisian* banks, we say

* These banks were open to the public and tothe municipal police. Of
the gaming-houses in London, those who know them must speak, The

ON THE RISKS OF LOSS OR GAIN. 111

that the permanency of each had 100 to 1 in its favour,
we are entitled to conclude that (the tenth root of 101
being 1°59) the games played were each not less than
3 to 2 in favour of the bank, if they staked the 10th
of their resources at each throw, or 11 to 10 if they
stake one fiftieth.

Prosiem. ‘The odds in favour of the bank being b
to a, required the greatest proportion of the fund which
may be staked at one game, in order to insure the chance
k to 1 for the permanency of the establishment.

Ruiz. Divide the logarithm of one more than k by
the’ excess of the logarithm of b over that of a: the
quotient is the denominator of the fraction required, 1
being the numerator. Suppose, for instance, that the
odds for the bank, on single games, are 30 to 29, then,
if 99 to 1 be required to be the chance that the bank
shall continue to exist, divide the logarithm of 99 + 1
(which is 2) by the excess of the logarithm of 30 (or
1:47712) over the logarithm of 29 (or 1:46240), and
2 —+-:01472 gives a fraction more than 135 ; whence
the 135th part of the capital is the highest which
should be staked.

A merchant, who engages in speculations which must
produce a fixed loss or a fixed gain, and who offers to
deal with any one in such a manner, is precisely in the
position of a bank such as is above described. The reason
why neither party need be ruined in this instance is
that the produce of the earth and sea is an unlimited fund,
upon which the merchant draws, Trade by itself would
tend to ruin the many, and accumulate all the stakes in
the hands of a few,and the theory of probabilities would
enable us to foretell that continual approximation to-
wards the extremes of wealth and poverty which com-
mercial countries always present. We have seen that
the poorer player must, to maintain his ground, have a

protection and encouragement which legal regulation of gambling-houses
would appear to give to gambling in general, is a goad reason for the state
of our own law ; otherwise, there can be no doubt that much particular evil
would be prevented.by allowing a regulated system.

112 ESSAY ON PROBABILITIES.

mathematical advantage in his favour. Now, it is the
nature of free trade that whatever mathematical ad-
vantage can be gained at all, is more accessible to the
rich speculator than to the poor one. Consequently,
the richer player, for that reason, can make himself the
stronger player. If, then, a certain number of persons
were to play upon a fixed total of stakes, equally divided
at the commencement, with the condition that every
stake won should enable the winner to make his next
throw with somewhat more (no matter how little) of
mathematical advantage than he had before, it is certain
that, in the long run, the whole of the stakes would be
in the hands of some one of the players. But, in the
actual state of things, there is always an accession of
new stakes and new players, so that the original players
are contending against an unlimited fund. If the con-
tinual augmentation of stakes and players be not suffi-
cient to counterbalance the tendency to extremes, a wise
government would throw the burthen of taxation more
upon the rich and less upon the poor. The mathe-
matical advantage of wealth would be taxed, as well as
its power of procuring luxuries. Such a result never
can be expected until the public mind is better informed
upon the subject of which this work treats.

CHAPTER VI

ON COMMON NOTIONS WITH REGARD TO PROBABILITY.

Tuose who have not considered this subject with par-
ticular attention, seldom fail to think that there must be
more or less of fallacy in the attempt to connect its prin-
ciples with its results. Some, indeed, of the latter are
strange and new, and are used as arguments against the
validity of the theory. JI propose in this chapter to

ON COMMON NOTIONS OF PROBABILITY. 113

turn those which precede to account, in examining
opinions of various kinds, whether on this subject at
large, or on particular cases of its application.

_ The doctrine of probabilities seems to some to assume
a sort of power of prophesying, or of predicting the
run of events ; to others, it appears that unless such a
power of prophesying be attained, the theory can be of
ho use. Both notions are correct in one sense and in-
correct in another: there is prophecy, but not of par-
ticular events, and derived, not from inspiration, but
from observation. The astronomer predicts—and all
the world knows that his predictions daily come true,
His means of prophecy are aided by deduction from
certain notions of which, be the cause what it may, we
are as certain as of our own existence. From his very
distinct (and therefore often called intuitive) pereeption,
that two straight lines cannot inclose a space, and various
other axioms of arithmetic and geometry, he is able to
make his observations tell him more as to the future
motions of our system than his unassisted perceptions
of the past could ever have accomplished. He is a
dealer in probabilities of a very high order. But be-
fore his prediction appears, it is necessary that he
should consider much more doubtful questions of pro-
bability. The minute errors of observation, coupled with
the various trifling effects which result from yet un-
discovered causes, oblige him to have recourse to the
principles which we have explained in the preceding
chapters.

Again, there is no prophecy of particular events in
the theory of probabilities, of which it is the very es-
sence that there should be more or less tendency to
falsehood in every one of its assertions. No result is
announced, except as having a certain chance in its
favour, which implies also a certain chance against it.

With regard to the second class of assertions, namely,
that unless the theory of probabilities enable us to
predict, it can be of no use —it may be said that, for
the purpose contemplated, it is of no use. Theory would

‘ :

114 ESSAY ON PROBABILITIES.

never enable us to tell what face of a die will be turned
up in any one instance, nor would the maxims of our
science be worth putting into practice with respect to
an event which is to happen only a few times. If a
man were determined to run six hazards, and never to
gamble afterwards, say if he were determined to wager
twice upon a pair of dice giving doublets, I should
think it perfectly immaterial whether he accepted an
even wager, namely 5 to 1, or not. For though, in the
long run, only one throw out of six will give doublets,
yet the probability that six throws will give such a pair
once at least is not very great. It is as a provider of
general rules of conduct that the science is valuable ;
the adherence to rules being desirable on precisely the
same principles as those which obtain in morals or legis-
lation, no maxim of which will be found to meet every
case which will occur.

It is an assumption of this theory that nothing ever
did happen, or ever will happen, without some particular
reason why it should have been precisely what it was,
and not any thing else. Conceive it possible that a ball
which is white might have been black, without the
alteration of any action or circumstance which took
place in time previous to the moment at which the
ball is shown, and the foundations of the theory of
probabilities have ceased to exist in the mind which has
formed that conception. There is no one but will admit,
that out of a box, which contains nothing but two bla@k
balls, nothing but black balls can be drawn; and that
out of a box which contains only two white balls,
no black balls can be drawn. The difficulty lies in a
clear perception of the remaining assertion; namely,
that when the box contains one white ball and one black
ball, a very large number of drawings will give as
many white as black nearly, and the more nearly the
greater the number. This proposition might be proved
in three ways: firstly, by actual experiment ; secondly,
by showing that out of all the possible cases which can
happen, those in which black and white are equal, or

ON COMMON NOTIONS OF PROBABILITY. 115

nearly equal, much exceed in number all the rest put
together ; thirdly, by showing that there can be no
possible reason for an excess of white, which does not
equally, by express condition of the question, apply in fa-
vour of an excess of black. The last is more unanswer-
able than convincing ; the second really shows that the
event which we propose and treat as one event, namely,
“as many white as black, or nearly so,” is, in fact, a
collection of a large number of events, much exceeding
in number all the rest which can happen. It is as if,
having a million of possible cases, I separated nine
hundred thousand from the rest, called each of them A,
and each of the rest B, and then asserted that A would
happen more often than B. But, nevertheless, I sus.
pect that to the first mode of demonstration, actual
experiment, most persons owe that degree of con-
fidence in the theory, which (often without knowing
it) they exhibit in the affairs of life; and I derive
such a suspicion from observing that every result of the
theory of probabilities which is not of a nature to admit
of every-day confirmation, or which would escape an
inattentive observer, is looked upon with distrust. In
no case is this more obvious than in the prevailing
notions with regard to luck.

It is observed that some people always have luck at
cards. The order of things seems disturbed at their
caprice ; if they sit opposite to the dealer at whist, then
there is always an undue proportion of trumps among
the cards which come second, sixth, tenth, &c., up to
the fiftieth; while, when they become before the dealer
— Hocus Pocus (for to no other spirit, ancient or
modern, can the agency be attributed) puts all the
good cards, third, seventh, eleventh, &c. The fact is
stated as a sort of mystery, and we hear of people
who are always lucky at cards and never at dice, or
vice versd. The statement implies that the parties who
make it believe there is something in luck — an asser-
tion which I do not think of questioning; for, as I

12 :

116 ESSAY ON PROBABILITIES.

shall proceed to show, it would be the most improbable
thing imaginable that there should be no such lucky
people.

Firstly, every question of probabilities stands in
precisely the same relation to our faculties, whether we
suppose a moral government of the universe, or none at
all, provided that we have no reason to suppose we know
any thing of the plan of that government in the par-
ticular case in question. If I am before an urn which
contains a black and a white ball, which is all I know,
I am then disposed to say the chances are even. The
ball which I am to draw is undetermined (by me), and
that which we call chance appears to exist. But suppose
1 draw the ball, and without looking at it hold it in my
hand. That which we call chance has ceased to exist —
the ball is actually determined, and I am clearly and
physically placed in the same position as I should have
had before the drawing, if a superintending power,
eapable of guiding my thoughts and actions without my ~
perceiving it, had predetermined which I should draw.
But my position with respect to knowledge of the ball is
not in any way changed, either by the predetermination
of the superintending power before the drawing or by
my own act of drawing, as long as I do not know what
I am to draw or have drawn. It tells me nothing, if I
hear that the drawing is settled, unless it be in a manner
by which I can form some guess as to the nature of the
settlement. Consequently we must not, unless some
reason be shown for it, consider the question of the
luck of individuals in any other light, with reference to
calculation, than that in which it would have been
placed by the supposition that, all imaginable species of
fortune being described on the tickets of a lottery, each
individual had one drawing made for him at birth,
which should describe his future successes and reverses.
To create an analogous question, within reasonable
numerical limits, let us suppose a thousand individuals,
each of whom is to play two thousand deals at

ON COMMON NOTIONS OF PROBABILITY. 117

whist, with a given suit as trumps.* Let there be a
lottery, containing an enormous number of books, in
each of which 2000 deals are described, and let the
books be so many in number that among them is one
containing every possible set of 2000 deals which can
be imagined, the four hands in each deal being described,
and that allotted to the drawer of the book being
marked as such. Let each individual draw one of these
books, replacing it before the next drawer arrives:
these individuals are then precisely in the same situation
with regard to us, if their hands are to be dealt to them
according to the directions laid down in their books, as
if the distribution were made by accident (as we call it).
Now the question is, which is most likely, that the luck
of these individuals shall all be nearly the same, or that
some of them shall have a marked predominance over
others? To take one simple question: consider only
the chance of gaining the ace of trumps. Excluding
the dealer's advantage, to simplify the question, the
chance of any one individual gaining the ace of trumps
at any given deal is 4. Considering 2000 deals, he
has a very good chance of gaining it 500 times, or a
few more or less. But the probabilities are much in
favour of several of these 1000 individuals having a
very different lot from the average. Frame a set of cir-
cumstances in this respect against which it shall be
twenty to one, and (page 43) itis a hundred to one that
this fate (ora better) shall be found to be that of some
one or other out of any 92 individuals taken at hazard
from among the thousand. And when to the chance of
holding the ace of trumps we add the various others
which constitute a good hand at the game, we thereby
much increase the probability of large fluctuations, one
way or the other; andthough itis certain that uniformity
will be found in the average lot of a large number of per-

* This does not alter the question ; since the substitution of four possible
different sorts of trumps would only multiply every possible case of good
and bad fortune four times.

13S

118 ESSAY ON PROBABILITIES.

sons, yet the larger the number, the greater will be the
extremes of fluctuation.

Now it must be noticed, that this variation is the
thing observed — not on one side only, but on both.
For every one who is luckyat cards, there is another who
is unlucky. It would be, indeed, such a sort of mystery
as that which 1 am endeavouring to explain, if the ex-
ceptions to common luck were all on one side, or if there
were no such thing at all as uncommon luck, or only in
very fewinstances. This latter would be the same sort
of phenomenon as we should see if a halfpenny gave
head and tail alternately through an enormous reeurrence
of throws. The event observed is precisely that which
might have been expected beforehand. If by thinking
mysteriously of the fluctuations of luck which are
observed in comparing the fortunes of individuals,
any reader should mean to imply that the alternative,
namely, slight individual departure from the average,
would not have been mysterious, he is in a singular
error, The state of things which he would regard with
no wonder would be an apparent interference with the
material world on the part of its governor, without the
intermediate agency of any second causes; that is,
something resembling a miracle. For though the phi-
losopher, in such a case, would suspect an intermediate
cause, and endeavour to discover it, this consideration
does not enter into the view of people in general. When
the world wonders, whether at one side or another of a
question of probabilities, it is at the want of any ap-
parent physical or moral reason: on which account they
refer it to the Creator in a manner different from that
in which they refer what they call usual occurrences.

The law of individual cases is, that there shall be
marked differences ; of the masses, that there shall be
great approach to uniformity. There are a hundred
years in which, and hundreds of diseases by which,
any individual who is born may die: a lottery, which
should contain one ticket for every disorder, repeated as
often as there are years of age in which it has been

ON COMMON NOTIONS OF PROBABILITY. 119

fatal, would present at least 20,000 chances. Before an
individual is born,it is, say 20,000 to 1 against his dying
at a given age of a given malady; and yet, even with
such imperfect observations as exist at present, it begins
to be seen that uniformity is the law of large masses
compared with each other. I will illustrate this by
some cases. Few things appear more varied than the
distribution of maladies, that of criminal acts, and that
of the sex of children in different families. I have
taken purposely a case of evil, physical and moral, and
one which is neither. The experience of any one
individual might lead him to say that it is no uncommon
thing for three or four times as many persons to die of
consumption in one period of five years as in the
previous period ; but the experience of one large city
will show that such is not the case. The bills of mor-
tality in London showed the following results; the
upper line denoting the last year of the five in question,
and the lower line the average number in every thousand

deaths which were caused by consumption, or what was
called such. *

PISZ OIG ST cs A eee eT ioe oe
135 163 165 180 187 197.

Here is nothing like enormous fluctuation. The
gradual increase of the number shows an increasing
tendency to the complaints then described under the
head consumption, but cannot be called fluctuation, being
itself regular.

The number of murders committed in the whole
extent of any one country might be supposed liable to
very large yearly fluctuation, and still more the compara-
tive numbers committed with different classes of weapons.
A few years ago, extreme derision would have followed
the assertion that the sword and pistol would be felo-

* The known loose manner in which these Tables were put together
does not affect the argument, further than to favour its conclusion. The

chances of error in the description increase the probability of fluctuation
in the results.
1 4

120 ESSAY ON PROBABILITIES.

niously used in different years by nearly the same num~
bers. Let us look at the following Table, extracted
from the Essai de Physique Sociale of M. Quetelet. The
country referred to is France.

|
veawe. 1826 | 1827/1828 | 1829/1830 1831)

wor Pa bine ertior liga 241 | 234 | 227 | 231 | 207 | 266
Fire-arms~ - - 56 64 60 61 57 88
Sharp weapons of

war 2 - 15 4 8 yf 12 30
Knives - - 2s 39 40 34 46 44 34
Clubs or sticks - 23 28 31 24 12 21
Stones - - 20 20 21 21 11 9}

Sharp instruments
not above de-
scribed - «| 85 40 | 42| 45] 46] 49

Striking or kicking | 28 | 12 | 21} 23] 17} 26

Other modes, and
unknown - -| 25} 23] 10 4 8 9

I now compare the number of male and female bap-
tisms registered in England in 1821, and the nine fol-
lowing years. For these successive years it is found
that for 1000 girls baptised there were 1048, 1047,
1047, 1041, 1049, 1046, 1047, 1043, 1043, and 1034
boys.

Such cases as the preceding tend to establish the law
in question ; namely, that different large masses of facts,
collected under the same circumstances, will present
nearly the same averages. I now proceed to another
point.

When two circumstances happen to change together,
it is frequently presumed that they are connected with
each other, when, in truth, there is no reason for any
such supposition. In order to justify any notion of

ON COMMON NOTIONS OF PROBABILITY. 12]

necessary connection it-is necessary that the two cir-
cumstances should always happen together, and that one
should never happen without the other. If it should
only be observed that one is very frequently accompanied
by the other, we must then inquire into the probability
that either may happen without the other. If two
events are almost always happening, then it is evident
that very frequent coincidence is no evidence of connec-
tion, so long as exceptions tell us that there is no neces~
sary connection. And even if we always observe A to
be immediately followed by B, it does not immediately
follow that they are necessarily connected. We must
remember that the phenomenon is this — our perception
of A is immediately followed by our perception of B.

This may arise in different ways, as follows.

1. A may make B necessarily follow it ; that is, the
common deduction from the phenomena may be true.

2. Our perception of A may make B follow. This
is the case with regard to all effects produced upon our
own minds by A.

3. A itself may make our perception of B follow.
' “he latter may be always happening, but it may require

he arrival of A to make us see it.

4. Our perception of A may make our perception of
B follow; that is, B may be always happening, but it
may need both the arrival of A and our knowledge of
that arrival to make us see B; or the circumstances
which favour our perception of A may be the same, or
have necessary connection with, those which do the same
for B.

5. It may be B which is the antecedent event ; but
our perception of A, the consequence of B, may be ne-
cessary to our perception of B itself.

We sometimes, for example, note what takes place
from the time A happens, and compare it with previous
events, merely because the arrival of A suggests the
comparison. In many instances we do this correctly,
and merely for convenience ; but whenever the events

122 ESSAY ON PROBABILITIES.

which are compared with A are such as to exhibit what
we call accidental fluctuations, we are apt to imagine
that the difference between the two sequences is a
consequence of A. In the mysterious subject of luck,
already alluded to, this tendency to error produces
superstition. There are many who imagine that the
change of seats, or a new pack of cards, changes the
luck. They imagine it, because they observe that
the luck is not the same before the change as after it,
which, for the most part, is true. But it is equally
true that, for the most part, no number of games exhi-
bits the same fortune as those which precede it; that
is, this change of luck is always taking place, but is
usually only perceived when the introduction of some
novel circumstance affords a point to date from, on one
side and the other. The growth of the superstition is
this: — An individual who has been unlucky during
several games, happens to begin to win after the intro-
duction of new cards. His fortune changes, as most
probably it will do; for if the chances be even, and
three games have previously been lost, it is seven to one
against the next three games resembling them, and an
even chance that he shall win the next game. If he
win the next, or, indeed, if he do not go on losing, he
notes the circumstance, and the next time a run of ill
luck occurs, he takes particular care to repeat the expes
riment. In this way he soon furnishes himself with a
tolerable number of facts in support of his theory.
The exceptions are forgotten ; for it is the character of
negative events to lay less firmly hold of the mind*
than positive ones. Thus the theory of the change of
the weather with that of the moon receives more con
firmation from one fact in its favour than of doubt from
two against it. This last notion is another case in
point, The weather of any three days, in by far the
most instances, differs from that of the preceding three

* The lucky hit of a prophet of the weather, in foretelling the coldest

day of January, 1838, did more to establish his infallibility than weeks of
succeeding mistakes could destroy.

ON COMMON NOTIONS OF PROBABILITY. 128

days. When once the notion is obtained that a change
of weather will follow that of the moon, the epoch is
watched, and the change which is in most instances
observed, is admitted as evidence. If any one would
carefully note for a considerable time the weather pre-
ceding and following prorogations of parliament, he
would perhaps astonish the world with a result which
no one has yet dreamt of. These cases fall under the
third head above described.

It was frequently supposed, a few years ago, that
comets produced hot weather. An examination of the
number of comets discovered in years of different
average temperature gave it as a result that there were
more comets in hot summers than in cold ones. But
since hot summers are generally fine, with clear skies,
and cold summers cloudy and rainy, it is obvious that
the former are more favourable to the discovery of
comets than the latter. The fact, then, from which the
inference was drawn amounted to this, that the years of
heat are those in which we see most comets. With
what we know of the matter, there is no more reason
to suppose that comets bring heat than that heat brings
comets. We must, in all instances of presumed con-
nection, lock clesely at these two distinct things — the
happening of an event, and our perception of it ; other-
wise, we shall always be liable to suppose that an event

may produce the first, when it produces only the second.
There is, however, a class of events which does not
appear capable of any of the preceding explanations ;
namely, the occurrence of what are called runs of luck
at play, and repetitions of similar events in common
life, which give rise to such proverbs as this — that it
never rains but it pours, and misfortunes never come
alone. We shall first attempt to destroy the extraor-
dinary character of these occurrences, and shall then
show that any other order of things would be indeed
extraordinary,

Let there be any event of an unusual character, say

124 ESSAY ON. PROBABILITIES.

the drawing of a black ball out of a lottery of twenty
balls in which only one is black.- Surely, any one would
say, we shall never draw the black ball fifty times
running. I answer, continue drawing as long as I
direct, and you certainly shall, that is, if you will admit
that an event must some time or other arrive, which has
ten thousand to one in its favour. Supposing the
ball to be replaced immediately after drawing, and the
lottery to be shaken, so that every ball has its fair chance,
I will calculate the number of drawings in which it is
much more than 10,000 to 1 that a black ball shall be
drawn five times running, and it will be evident that by
the same principles the number might be calculated
which would give as great odds for a run of 50, or
5000, or any number, however great.

The chance of drawing the black ball at oe a is
#0 , and ao five shea trials it is =, x 35 X
sh X ab X ob UW azosa7G Itis, therefore, 3, 199,999
to 1, say 3,200,000 to 1, against five successive black
balls. In page 43, look in the Table opposite to 10,000
to 1, and we see 921. Let every succession of five
drawings be called a set, then it is 3,200,000 to 1 against
any given set being all black. Multiply 3,200,000 by
921, and divide by 100, which gives 29,472,000, and
this is the number of sets in which it is 10,000 to 1
that one whole set, at least, shall be black. Much
greater are the odds that there shall be a run of five
somewhere in all the drawings which make up these
sets ; for this latter event would arrive, not only when
all of a set are black, but when the last of one set, and
four of the next, or the two last of one set and three of
the next, &c. are black.

Let two players, with equal chances, play a large
number of games in succession. It is 63 to 1 against
a run of six games in one given manner, and 62 to
2,or 31 to 1 against arun of six games for one or
the other. Now, 31 x 921~—100=285°'51, or it is,
as before, more than 10,000 to 1 that in 286 sets of
six, arun of six shall occur for one or theother. This

ON COMMON NOTIONS OF PROBABILITY. 12s

vives 1716 games, and yet a sturdy whist player, who
plays on an average a dozen deals an evening for 200
evenings in the year, or 2400 deals per annum, will look
grave when he relates that he had bad cards for six
deals together, and will assure you that there is some-
thing in luck.

But at the same time, the number of trials which
makes a run of the unlikely event extremely probable,
will give the same probability to a much larger run in
favour of the more probable event. ‘To find to what
extent this goes, use the following

Rute. If it be a tod for A against B, in any single
trial, then subtract the logarithms of a and 6 separately
from that of a + 6; take any two whole numbers
which are very nearly in proportion to these differences,
and to each add 1; the sums show the runs of the
two events, which have the same probability (be it small
or great) of happening in a very large number of
throws. Thus, suppose it is 10 to 3 for A against B.

Log, 13 111394 log. 13 1°11394
Log. 10 1:00000 log. 3 °47712

0°11394 0°63682 :: 11: 64 nearly.

Consequently, whatever chance there is for arun of 12
B’s, there is as much for a run of 65 A’s.

In the last mentioned peed when a = 19, 6 = 1,
we have log 20 —log 19 = :02228, and log 20—log t
a= ?*30103;, which differences are 1 to 65 nearly, and
as 6 to 390. Consequently there is as much reason to
expect a run of 391 white balls as of 7 black balls.

No person can take a rational view of probabilities
until he ceases to recoil from the supposition that an
event is never to happen because the odds are very much
against his choosing, out of a large number of trials,
the one in which itis to happen. The best way to force
the mind upon the consideration is to return to the first

\

126 ESSAY ON PROBABILITIES.

principles upon which the method of judging is founded.
You find it difficult to imagine, that out of twenty balls,
one only of which is black, you shall draw the black
hall five times running. But yet in 30,000,000 sets of
five drawings each, it is asserted that you are what is
called “ almost sure” of drawing the black ball through-
out the whole of one set. Waiving the question of
probabilities, I will now state what it is of which ma-_
thematical demonstration makes us quite sure. Let
vol. i. be a book which describes 30,000,000 of sets ;
vol. ii. another, which describes 30,000,000 more, differ-
ing from the preceding in some, many, or all, its sets,
and so on until every possible collection of 30,000,000
of sets is described in one volume or another. Now it is
quite certain that out of the mnumerable volumes which
will thus be produced, the volumes which somewhere or
other describe a set all black will outnumber those which
do not describe such a set many thousand times, 10,000
at least. Suppose the black sets when they exist, to be in
a frontispiece; the question then is, having an enormous
library, with books at the rate of 10,000 with a frontis.
piece, for one which has none, and taking down a book
blindfold, which do you suppose to be most likely, that
you shall draw a frontispiece, or none at all? Un-
questionably you answer that you are almost mathema-
tically certain of not drawing the latter. But this is
(page 124) an exaggerated statement of the case of the
chance of a run of five black balls in 30,000,000 of sets.
But it is said that no person ever does arrive at such
extremely improbable eases as the one just cited. That
a given individual should never throw an ace twelve
times running on a single die, is by far the most likely ;
indeed, so remote are the chances of such an event in
any twelve trials(more than 2000,000,000 to 1 againstit),
that it is unlikely the experience of any given country,
in any given century, should furnish it. But let us
stop for a moment, and ask ourselves to what this argu-
ment applies. A person who rarely touches dice will
hardly believe that doublets sometimes occur three times

ON COMMON NOTIONS OF PROBABILITY. IZ?

running ; one who handles them frequently knows that
such is sometimes the fact. Every very practised user
of those implements has seen still rarer sequences. Now
suppose that a society of persons had thrown the dice so
often as to secure a run of six aces observed and recorded,
the preceding argument would still be used against
twelve. And if another society had practised long
enough to see twelve aces following each other, they
might still employ the same method of doubting as to
arun of twenty-four, and so on, ad infinitum. The
power of imagining cases which contain long combin-
ations so much exceeds that of exhibiting and arranging
them, that it is easy to assign a telegraph which should
make a separate signal for every grain of sand in a globe
as large as the visible universe, upon the hypothesis of
the most space-penetrating astronomer. The fallacy of
the preceding objection lies in supposing events in
number beyond our experience, composed entirely of
sequences such as fall within our experience. It makes
the past necessarily contain the whole, as to the quality
of its components ; and judges by samples. Now the
least cautious buyer of grain requires to examine a
handful before he judges of a bushel, and a bushel be-
fore he judges of a load. But relatively to such enor-
mous numbers of combinations as are frequently pro-
posed, our experience does not deserve the title of a
handful as compared with a bushel, or even of a single
grain.

Further to illustrate this point, let us turn back to
page 04, where we see that when an event has happened
m times running, it is m + 1 to m that it shall happen
n times more. ‘This proceeds upon the supposition that
the chances of the event were entirely unknown before
its happening, and the presumptions drawn are therefore
entirely derived from experience. When an event has
happened 1000 times one way, it is 1001 to 1 that it
happens in the same way next time. But itis only
1001 to 1000 that it happens 1000 times more the
same way, and only 1001 to 1,000,000 that it happens

128 ESSAY ON PROBABILITIES.

1,000,000 of times more in the same way. Hence ex-
perience can never, on sound principles, be held as fore-
telling all that is to come. The order of things, the
laws of nature, and all those phrases by which we try to
make the past command the future, will be understood
by a person who admits the principles of which I treat
as of limited application, not giving the highest degree of
probability to more than-a definite and limited continu-
ance of those things which appear to us most stable. Vo
finite experience whatsoever can justify us in saying that
the future shall coincide with the past in all time to come,
or that there is any probability for such a conclusion.

CHAPTER VII.

ON ERRORS OF OBSERVATION, AND RISKS OF MISTAKE.

In every measurement, as well as in unassisted esti-
mation, the observer is liable to error; the only dif-
ference being that the mistakes of careful instrumental
measurement are likely to be less than those of esti-
mation. That which we call estimation means guess
formed by a person whose previous habits and ex-
perience are such as to make it very likely that he can
tell nearly true that which would require instruments
to obtain with great approach to accuracy. To illustrate
this distinction, imagine a certain small length, say
about twelve inches, to be presented to a large number
of persons, who are required to write on separate bits
of paper how many inches and tenths of inches it
appears to them to contain. If these persons had been
used to estimate lengths by the unassisted eye, it would
be extremely probable, 1. That the average of their
guesses would be very near the truth. 2. That their
widest limits of error would be small. If their habits

ON ERRORS OF OBSERVATION. 129

have not been accurate, it is still reasonably probable
that the average of their guesses would be nearly true ;
the limits of error would certainly be larger. It is the
object of the present chapter to show how the theory of
probabilities must be applied to the detection of the
most probable result, when various observations are
discordant with each other.

Error, as used in this part of the subject, merely
means discordance of which the cause is unknown. In
the different branches of physics, in their application to
the arts, &c. &c., that which we signify by the pre-
ceding word may arise from various causes, the chief of
which are here enumerated.

1. From some law of nature not known to the
observer. ‘Thus before the discovery of the aberration
of light, all the small yearly changes which the places
of stars receive from that cause, only appeared in the
form of embarrassing differences between the observ-
ations of different months. Those who used astro-
nomical instruments might suspect the existence of
some unknown motion in the heavenly bodies; they
might think it extremely improbable that their improve-
ments in the art of observing should permit purely
casual discrepancies of so large an amount as those
which occurred : but still, so long as no account of the
magnitude of these possible results of law could be
given, those who observed could in no manner dis-
tinguish them from the imperfections of the instru-
ments, or of the human senses. But had it been
shown that these discrepancies were always the same at
the same time of the year, for any one star, they would
then have ceased to be errors, and would have become
the objects of prediction, as soon as one year’s observ-
ations had been registered. The physical cause might
or might not have been subsequently discovered, without
altering the state of the question: the certainty of a
phenomenon is all that is required to remove it from
the domain of probability.

2. From the personal constitution of the observer ;

K

130 ESSAY ON PROBABILITIES.

by which is meant, not that general facility of misper-
ception which is common to all the human race, but
that particular habit or temperament Which causes some
to differ from persons in general in their method of
perceiving. Thus it will frequently happen that when
two observers note the time of a phenomenon, by the
same watch, one will always see the event, or imagine
he sees it, before the other does the same. This
personal error, which is seldom large, is beginning to
receive the attention of observers. It is not perceptible
as long as the natural data of a science remain imper-
fectly known, being mixed up and lost in errors of
greater magnitude; but it produces discoverable effects
so soon as the science approaches towards accuracy.

8. From fixed sources of error peculiar either to
the species of apparatus employed, or to the individual
instrument with which the observations are made.
This answers precisely to the personal error of the
observer in its effects: it matters nothing whether the
clock be one second too fast, or the observer, in the
result of the observation.

4. From the imperfection of human senses and in-
struments. ‘To note a measurable phenomenon without
any error at all, would require sight and touch by
which every magnitude, however small, could be per-
ceived and correctly estimated. Such senses belong to
no one, and the degree of approach towards perfection
not only varies with the observer, but is different at
different times with the same observer. Many errors
to which instruments are subject ought in strictness to
be classed under the first head ;.if, for instance, an
astronomical circle gradually change its form, or
undergo daily expansion and contraction by variations
of temperature, the diversity of results which such a
piece of brass will shew are certainly subject to laws,
and might be predicted, if we possessed sufficient
knowledge of the constitution of the metal, and the
laws which regulate the effect of pressure, temperature,
moisture, &c. upon it, But so long as such laws are

2
an
i

ON ERRORS OF OBSERVATION. 131

unknown, and the variations do not follow any dis-
tinguishable rule, their effect upon general results
differs in nothing perceptible from that of the observer’s
own errors, with which they are mixed up in the par-
ticular results of observation.

Before any trials are made, that is, before any thing
is known of the character of the observer, of the instru-
ment he uses, or the perceptibility of the phenomenon,
we can have no reason to suppose that any one observ-
ation is more likely to exceed the truth than to fall
short of it. When any observation is greater than the
reality, the error is called positive ; when less, negative.
The hypothesis, therefore, of an equal presumption for
positive and negative errors, is one with which we must
commence ; and it follows from the supposition that
the average is the most probable result of a number of
discordant observations. The sum of all the observations
will be affected by the balance of all the errors, but
will be without error itself if the amount of the positive
errors be equal to that of the negative ones. This last
supposition, though not probable in itself, is never-
theless more probable than any other, and the odds are
very much in favour of its being nearly true. Now
whatever may be the error of the sum of observations,
say 100 in number, the average, or the hundredth part
of the sum, contains only the hundredth part of that
error; and the presumption that such an average is
very near indeed to the truth, greatly exceeds the pro-
bability in favour of any one of the observations.

But before we proceed further, it becomes necessary
to ask what laws of error can be absolutely determined,
and shown in the nature of things to exist? And
first, what do we mean by a law of error? Let AB
be a length to be measured * or estimated, subject to
error of observers and instruments ; and let the greatest
possible errors be B K (negative), and B L (positive) ;
so that the result of measurement may be any thing

* The second figure is an enlargement of part of the first.

K 2

132 ESSAY ON PROBABILITIES.

between A K and AL. Let K B and BL be equal,
and suppose positive and negative errors equally likely

a ei io
>,

1 poee?

A K . t,

The probability of any one measurement giving
exactly any predicted result, say A N, must be in-
appreciably small; since AN is only one out of an
infinite number of possible cases. But take any point
V, however near to N, and the chance that the result

Cc

a.

K w B NV L

of an observation shall lie between AN and AV is
capable of being imagined to be finite, though small.
Whatever the law of errors may be, let a curve KCL
be so drawn that the chance of something between
AN and AV shall be the proportion which the area
NPQYV is of the whole area KCL. Or if we call the
whole area 1, the area RPNW, for example, is that
fraction which expresses the chance of a result lying
between AW and AV. ‘The symmetry of the curve
on the two sides of CB is an expression of the hypo-
thesis of positive and negative errors being equally
likely : and the approach of CK and CL towards the
axis is equally an expression of the supposition that
large errors are not so likely as small ones. If we wish

ON ERRORS OF OBSERVATION. 133

to express that errors of all magnitudes are very nearly
equally likely, we draw such a curve as the upper
dotted line: while, if we wish to express that the pre-
sumptions are very strong for each measure being very
nearly true, we describe the lower dotted line. We
can thus figure to the eye a representation of the law of
errors, be it what it may ; and the description of a law of
error is that of a curve.

Generally speaking, it is impossible to commit an
error of more than a certain magnitude: but this cir-
cumstance is one which embarrasses the mathematical
treatment of the subject exceedingly. It is practically
the same thing to consider any error, however great, as
possible, but errors of more than a certain magnitude
as extremely improbable. If, for instance, a case should
arise in which an error of more than an inch is impos-
sible, let it be agreed to consider such an error as not
impossible, but so improbable that there shall not be
an even chance of its happening once in a million of
times. The curve which exhibits the law of error
must then be of the following kind, never meeting the
axis, but continually approaching towards it, so that the
whole of its area from and after L, is incomparably
small by the side of the area CBL.

C

K B L

The curve here drawn is something like that in
p- 17., and we might suspect from the utility of the
latter and of the tables derived from it, that it should
play an important part in the present subject. This
we shall presently see: in the mean while I proceed to
explain the sense in which I use the terms average
balance, average error, mean risk of error and probable
error, to which I direct the reader’s particular at.

Kk 3

134 ESSAY ON PROBABILITIES.

tention. When we consider the errors of different
kinds as balancing each other, it follows that positive
and negative errors being equally likely, the balance of
all the errors will be trivial in a very great number of
observations. The average of all the errors will be
extremely small: and, in the long run, nothing. In
this instance then, it is said that the average balance of
error is nothing. But if positive and negative errors
were not equally likely, the more probable class would,
in the long run, predominate, and the average balance
of error would be of a definite magnitude, positive when
positive error is more likely than negative, and vice
versa. |

_ The preceding has nothing to do with the average
of the absolute magnitudes of errors, considered without
reference to the distinction of positive and negative. For
example, whether the greatest error may be a mile or an
inch, it is equally true that the long run will establish
a compensation, when positive and negative errors are
equally likely. But in the former case, the average
magnitude of the errors which occur will, ceteris paribus,
as much exceed that in the latter, as a mile does an
inch. ‘This average magnitude of errors, independently
of sign, is an important element of the whole question,
because a tolerably probable estimation of its value can
be found from the observations. Suppose, for instance,
that fifteen observations or estimations gave the follow-
ing results.*

722 13811 967 _1809
933 1089 1344 858

1033. 972 1250 1029
917 1294 ‘744

The average of these is 1051, and assuming this as
the true result, the errors are,
329 260 84 258
118 38 . 293 193

18 79° «199 22
194: 243... 807

* These are not numbers written at hazard, but poo results ot
estimation, on a subject which it is not here necessary to explain.

ON ERRORS OF OBSERVATION. 135

The average of the errors is 172 ; subject of course to
the supposition that the average of the observations is
nearly true. But the error of the last assumption
must be considerable before it can much affect the
present result.

The average of all the errors, taken without reference
to sign, will, in the present hypothesis, be the same
thing in the long run as the average of positive error,
or the average of negative error. The reason is, that
the number of positive and negative errors will in the
long run be equal, and also their sums. If, in a very
large number of observations, there be s positive and
s negative errors, and if the sum of each set be §, it
is evident that the sth part of S (which is the average
positive error, or the average negative error), is the
same as the 2sth part of 2S (which is the average
error without reference to sign). But it is customary
to prefer to the average error another function of the
errors, which may be called the mean risk of error, and
which differs from the average in the following manner.
Every error, positive or negative, is an increase or
diminution of the final result; just as every game
won or lost at gambling is an increase or diminution
of the stock of the player. On precisely the same
principles as those explained in chapter V, I may con-
sider an even chance of an error 2 as a thing to be
compounded for by a certainty of an error 1. If, in a
large number of observations 2s, there will come a sum
of positive errors equal to S, and the same of negative
errors, and if, as in taking the average, the sum of the
errors be the only material point, it may be considered
that every observation will have either a positive error or
a negative error, of the value of the sth part of S. The
results of such a supposition will, in the long run, and so
far as the sum of errors is concerned, represent the actual
case under consideration. If, therefore, a person could
compound for positive errors alone, leaving the negative
ones to chance, he must suppose every observation to
have the half of S = s, or S + 2s of positive error,

K 4

136 ESSAY ON PROBABILITIES.

combined with such a negative error as chance may
yield. That is, he must suppose all the negative
errors altered by the introduction of such an additional
positive error, and each of the positive errors increased
or reduced to S+2s. And the same if he would
compound for negative errors only: while, to compound
for both, he must suppose every observation affected
both by a positive error of S + 2s and by a negative
error of the same amount. This latter case supposes
every observation to be correct, which is the result in
the long run. The use of this consideration is, to keep
before the mind the average effect of positive error, not
upon those observations which have positive error, but
upon all the observations ; and the same for negative
errors. Suppose I have an instrument which makes
positive and negative errors in equal numbers and to
equal amounts, in the long run ; and suppose that it is
in my power totally to destroy positive error, leaving
the chance of negative error as before. What is the
effect upon the whole system of errors, positive and
negative, one with another? What must I do with all
the errors to reproduce the same amount of absolute
error as before? I must affect every observation with
one half of the average amount of positive error, or one
half the magnitude which the positive errors have, one
with another. Or look at it in this way ; if I have to
pay a shilling for every unit of positive error, for how
much should another person take the risk off my hands,
that is, for how much per observation, whether its
error be positive or negative. If the average positive
error were 1, I should have in the long run to pay at
the rate of one shilling for every two observations, against
which I should insure at the rate of sixpence per
observation. ,

_ The mean risk of positive error, then, is the average
positive error, when the errors of this kind are equally
distributed over ali the observations: and the same for
negative error. When positive and negative error are

ON ERRORS OF OBSERVATION. 137

equally likely, each is one half the average error, con-
sidered without reference to sign.

By the probable error I mean that amount of error
which is such, that there is an even chance for exceed-
ing or falling short of it. Thusif it be 1 to 1 that
the error shall lie between O and 10, and of course
the same that it shall exceed 10, then 10 is called the
probable error. For any number greater than 10, the
chances are (no matter how little) in favour of the
error being within that number; for any thing less
than 10, the chances are against the error falling
within that amount.

Prositem. The number of observations being n,
and positive and negative errors being equally likely,
required the probability that the average of the n
observations lies within a given quantity & of the truth ;
or, M being the average, that the truth lies within
M +k and M —k. |

Rute. (By Table I.) Take the average of the
observations, find all the errors upon the supposition
that the average is the true result, add together the
squares of the errors, and divide the square of the
number of observations by twice the sum of the squares
of the errors. Call the result * the weight of the
average. Multiply & by the square root of the weight ;
let the result be t; then the H answering to t in table
I, is the probability required.

Ruts. (By Table II.) Find the weight as in the
last rule, and divide 62 by 130 times the square root
of the weight. The result is the probable error of the
average. Divide & by the probable error, and let the
quotient be t; then the x answering to t in table II. is
the probability required.

Exampxe. In the preceding instance, what is the
probability that the average 1051 lies within 50 of the
truth. The squares of the errors are, 108241, 13924,
324, 17956, 67600, 1444, 6241, 59049, 7056,

* The reason of the appellation will be afterwards explained.

138 ESSAY ON PROBABILITIES,

85849, 39601, 94249, 66564, 37249, 484, the sum
of which is 605831; and twice this is 1211662.
Divide 225, the square of 15, by 1211662, which gives
0001856953, the weight of the average. The square
root of the weight is ‘013627, which multiplied by 50
gives 6814, the value of t: that of H is then -66.
So that it is more than 3 to 2 that the true result of
the preceding very discordant observations lies between
1001 and 1101

To use the second table, multiply -013627 by 130
which gives 1°77151, by which divide 62, giving 35
very nearly. ‘This is the probable error, so that it is
an even chance the result lies between 1051 — 35 and
1051 + 35, or 1016 and 1086. Divide 50 by 35
giving 1°43 the value of t; for which in table II., K
is ‘66, as before.

I now proceed to explain the meaning of the term
weight, as used above. When an observer has made
various observations, one or more of which he thinks
superior to the rest, as to the favourableness of the
circumstances under which they were made, it follows
that the good observations should ¢el/ more in the
formation of the most probable result than the indif-
ferent ones. If for example, a remarkably good trial
give 10 and an indifferent one 11, it is not reasonable
to say that 105 is the most probable result. If the
first observation be remarkably good, it may seem not
unfair to give it the force of four observations, or to
let the number 10 have the weight which would result
from four observations giving 10, 10, 10, 10, a fifth
giving 11. On this supposition the average is the
fifth part of 51, or 104, instead of 101. This was called
giving the observations 10 and 11 weights of 4 and 1,
and the method of finding an average is this: multiply
every observation by its weight and divide the sum of
the products by the sum of the weights. Such a method
was adopted before the theory of probabilities was ap-
plied to the subject, as a direction of common sense.
When that theory came into use, it was found that the

ON ERRORS OF OBSERVATION. 139

square of the number of observations divided by twice
the sum of the squares of the reputed errors (the
average being reputed correct) ought to stand in the
place of the weight in the preceding rule, whenever
different averages are to be combined together to form
one general average. If, for instance, one average of
100 observations gave 10 and another of 50 gave 11,
and if the squares of 100 and 50 respectively divided
by twice the sums of the squares of the errors gave 1°5
and 1:1, the most probable average of these averages
would not be 104, but the product of 10 and 1°5
increased by that of 11 and 1:1, and divided by the
sum of 1°5 and 1°1, which gives 10°4. Hence the
term weight is now applied to the quotient above de-
scribed.

When the law of error is of the kind figured in
p- 17., the mean risk of either sort of error, the probable
error, and the weight of a single observation, are con-
nected by very simple relations, as follows :—

1. The mean risk is 200 divided by 709 times the
square root of the weight; more nearly, :2820953
divided by the square root of the weight.

2. The probable error is 62 divided by 130 times
the square root of the weight ; more nearly, -476936
divided by the square root of the weight.

8. The weight is 113 divided by 1420 times the
square of the mean risk.

4. The probable error is 1-',; of the mean risk ;
more nearly 1°690694 of the mean risk.

5. The weight is 5 divided by 22 times the square
of the probable error ; more nearly, -227468 divided
by the square of the probable error.

6. The mean risk is 12 of the probable error ; more
nearly, ‘591473 multiplied by the probable error.

We can thus find the remaining two, when either
of the three is given. Of the three, I apprehend that
the probable error refers to the most instructive notion ;
but the mean risk and the specific weight enter more
usefully into formule .of calculation. The average

140 ESSAY ON PROBABILITIES.

error, being twice the mean risk, is readily determined
when wanted.

Many persons confuse the average and the probable
error in their own minds: that is, they imagine it to
be as likely that any error should exceed the average
as fall short of it. That such cannot be the case is
evident from the following consideration.

The average error depends upon the magnitude of
the error, as well as upon the proportions in which
errors of different magnitudes enter; the probable error
depends only upon the latter. If, then, small errors
enter in larger numbers than great ones, the probable
error is rendered less than it would otherwise be. In
determining the probable error, the error 100 entering
once, counts no more than the error 1 entering once.
But in the average error, the error 100, entering once,
counts 100 times as much as the error 1 entering once.
Consequently the former must be less than the latter.
But whether the probable error exceed or fall short of
the mean risk (half the average error), must depend on
the law of error. In the present case the former con-
siderably exceeds the latter.

It may be asked whether the preceding results are
always strictly true. Granting that the probability of
an error diminishes with its magnitude, and _ that
positive and negative errors are equally likely (which
are the only hypotheses of the preceding question), does
it necessarily follow, whatever may be the law of the
diminution of probability, that the mean risk of error
will, in the long run, be +42 of that error which is
as often exceeded as not? The absolute answer to this
is, that the assertion is not strictly true, except upon
further suppositions as to the law of error. The
manner in which this inconsistency is explained, depends
upon whether the person asking the question be sup-
posed to be an inquirer seeking methods of disciplining
his judgment, or an experimental philosopher requiring
only a sufficient practical rule for the treatment of a
set of observations.

ON ERRORS OF OBSERVATION. 141

To one who is looking for sound principles, I observe
that he does not want in this matter the exposition of
the consequences of any one law of facility of error,
but an account of the general character of those laws
to which common sense and daily experience assure
him that his faculties and means of observation are
subject. The facts of which he stands assured are,
that the probability of error does not diminish very
rapidly at first, but that as the error we consider grows
larger, its probability does diminish very rapidly, and
becomes insensibly small for errors of a certain magni-
tude and upwards. No curve of comparison, drawn in
the manner described in p. 132., will be a true repre-
sentation of what we know on this subject, unless it
have the general form, of which the following varieties
are instances. Now though the preceding results are
not strictly true for every curve which has such a form,
yet there is a class of curves of this form, some variety
or other of which will approach tolerably close to any
line which can be drawn to resemble one of those in the

Ne

sk cade

Nema ee ee Te RT EE

figure. In every one of this standard class of curves,
all the preceding relations are strictly true, and there-
fore are nearly true for all the curves which resemble .
any one of the standard class. Thus though the
average error may not always be +9 of the probable
_ error, yet the former is always some fraction of the

latter, not differing very greatly from 149. There
is another reason for the adoption of this law of error
as a standard, for which the reader may consult the.
fourth appendix to this work.

142 ESSAY ON PROBABILITIES.

The experimenter, looking for a method of treating
observations which shall produce trustworthy results,
well knows that it matters nothing whether a method
be true or false, if demonstration can be given
that the consequences of the method are true. That
falsehood necessarily produces falsehood is a fallacy,
pardonable enough in everything but mathematics.
True reasoning on true hypotheses must necessarily
produce true results ; false reasoning, or false principles,
or both, may, and most probably will, lead to false con-
sequences, but may lead to the direct reverse. In
every part of knowledge, except mathematics, error
must be carefully avoided, because there is no method
of distinguishing between the cases in which it leads to
truth, and the contrary cases. But in the exact
sciences, the knowledge of the consequences of false-
hood and of those of truth are equally evact: and it is
possible to introduce an erroneous addition to the con-
ditions of a problem, to trace the consequences of such
error, and to annihilate them at any part of the process.
It is possible also to substitute for truth an erroneous
supposition, in such manner that the effect of successive
lapses of this kind shall be compensatory of each other,
or so that the more often the error is repeated, the nearer
is the result tothe truth. The preceding case affords an
instance: let the law of error be what it may (provided
only that positive and negative errors are equally likely
and that of two errors the larger is always the less
probable), and let a moderately large number of observa-
tions be in question, and it follows that the results of the
real law, and those of the preceding supposition, are
nearly identical. Let the number of observations be
still larger, and the resemblance is still nearer, and so
on without limit. And this is true, even when the
law of error, as regards a single observation, or two or
three observations, varies to a large amount from that
which is used above. Consequently, for a tolerable
number of observations, it is absolutely indifferent
whether the real law of error be known, or whether

ON ERRORS OF OBSERVATION. 143

the nearest variety of the class under consideration be
substituted for it.

Having then asserted, as a result of investigation, the
existence of a standard law of facility of error which
not only represents or resembles the impressions which
unassisted reason would form 4 priori, but the results of
which are more than sufficient mathematical approxi-
mations to truth, whatever (with some easily admissible
limitations) the law of error may be, I proceed to
describe it more particularly, calling it in future the
standard law of facility of error. The sole datum
necessary for its specific application, is either of the
three, the weight of an observation, the mean risk of
error, and the probable error, any two of which may be
deduced from the third by the rules in p. 137.

Prosiem. Given either of the three data, required
—¥(A) the chance that the error of any one observation
shall lie between e positive and e negative, or that the
observation shall give something between e too much,
and e too littk—(B) the chance that the preceding
shall not happen—(C) the chance that the error shall
be positive, but not exceeding e; or that it shall be
negative, not exceeding e.

Rutr. Multiply e by the square root of the weight,
and let the product be t; then (A) is the H corres-
ponding to t in table I., and (B) is the remainder of (A)
subtracted from unity ; each of the chances called (C)
is one half of (A).

Exampie. The mean risk of error is 10 ; required the
chance of the error lying between + 15 and—15 ; that
is, between 15 too much, and 15 too little. Since the
square root of the weight is 200 divided by 709 times
the mean risk of error 10, or ;*,9;, and since 15 times this
result is #§ or ‘42; the probability required is °45 ;
or 11 to 9 against the event. The probability of a
positive error less than 15 is ‘225, and the same for a
negative error within the same limits.

When all the individual observations are made under
the same circumstances, so as to have the same weight,

144 ESSAY ON PROBABILITIES.

the common average is the most probable truth, and its
weight is as many times the weight of one observation as
there are observations in all: or, the average of n
observations each of which has the weight w, is entitled
to the same confidence as one observation made under
circumstances which give it a weight nw. Of this we
have an example in p. 137., and I now give another,
detached from the method there given of finding the
weight.

Exampire. The weight of each observation being
18, what is the probability that the average of 50
observations lies within °05 of the truth. The weight
of this average is 18 x 50 or 900, the square root
of which is 30. Now :05 x 30 is 1°5, which, being t,
H is -966, so that the probability required is about 23 to
1 in favour of the event. The mean risk of error of such -
an average is 299 divided by 30, or >4$-, less than
°01 ; by which it is meant that if repeated sets of 50
observations each were made, the errors of these sets,
neglecting their signs, would not average so much as
°01 x 2 or :02 each.

Let us now suppose that positive and negative errors
are not equally likely. Hitherto, the absolute truth
has been the most likely ; that is, though the pro-
bability of any one observation giving mathematical
truth was infinitely small, yet so was that of any given
error being exactly attained, and the infinitely small
probability of the first case was greater than that of
the second.* Let us now suppose that errors equal to
or near to P are more probable than any other, and so
that, 2 being the truth, any observation is equally likely
to exceed or to fall short of (not #) buta+P. This
is equivalent to describing a curve of the following
figure in place of that in p.132. Here AB is to be

* To compare the proportions of these indefinitely small probabilities,
say of absolute truth and of either the error +e, or that of —e exactly,
take H/ from table L, corresponding to 0, and to e multiplied by the square
root ofc. Thus, c being 1, the probabilities of an error 0 and an error 2
pre as 1‘1 to ‘02, or as 55 tol.

ON ERRORS OF OBSERVATION. 145

Cc.
ad bitin,

ee oo es

measured, BK is P, and the chance of any error
failing between O and BN is such a fraction of unity as
the area BPN is of the whole area of the curve. The
case before us is precisely that of an observer with a
personal error equal to P, in addition to casualties. If
we imagine a very large number of observations, made
under circumstances equally favourable to positive and
negative error, and if the error P be added to or taken
away from each of the results, according as P is a
positive or a negative error, we shall then have such a
succession of results as might be looked for on the
present hypothesis. For instance, let the quantity P
be 10, then whatever may be the prospect of having
the error 1, when positive and negative errors are
equally likely, we have the same chance for the error
{1, in the present case.

For simplification, I shall adopt the algebraical
method of signifying positive and negative errors.
Thus + 3 means 3 too much, or 3 added to the truth ;
— 4 means 4 too little. Thus +4—5is—1; mean-
ing that four too much from one cause, and 5 too little
from another, gives on the whole 1 too little. Again,
we consider — 2 less than — 1, since the effect of the
former is to lessen the result more than would be done
by the latter. To show that the supposition now before
us is equivalent to that of an observer with a personal
error, or an instrument with an individual error, imagine’
an instrument wrongly graduated, so that for every
reading we ought to read 10 less: thus for 125 we
ought to read 115. In other respects let the instru-
ment be equally likely to give positive and negative
departures from truth. If then an observation give
176 we know immediately that the truth is 176-— 10

L

146 ESSAY ON PROBABILITIES.

+a casual error, the effect of which disappears in a
large number of observations. The whole error then
is 10 + the casual error, 10 + # and 10 — & being equally
likely. This is precisely the hypothesis in question,
in the case in which P = 10.

The average of observations, in the case before us,
does not necessarily give an approximation to the truth.
Calling the quantity P a fixed error, meaning by a fixed
error one which is as likely to be exceeded as not, we
see, in the following theorem, a justification of the term :
the most probable result of a large number of observations
is the truth, increased or diminished by the fixed error,
according as it is positive or negative. This error may
either be fixed in the instrument, fixed in the observer,
or in different degrees in both.

Let the phenomenon to be observed be perfectly
unknown, except by what the instrument tells us.
Then it is totally impossible to discover the amount of
this error ; which, nevertheless, must be assumed to
exist, unless the contrary be shewn. For example, an
instrument wrongly graduated throughout could never
tell the truth, either in individual or average results.
But it is obvious that such an apparatus, incapable as
it is of telling absolute truths, might nevertheless detect
such results as are obtained by measuring the differences
of other results. Thus a clock which goes truly, but
is set too fast or too slow, will serve to find the time
elapsed between two events, though it will not show the
real time of either. Instruments which, on account of
some permanent error affecting all their results, can
only be used to determine differences, are called dif-
ferential instruments.

All instruments, as well as observers, are subject
more or less to this species of error; how then is it
possible to depend upon the results of any observations ?
The answer to this question will require some detail.
Since perfect exactness cannot be attained, either on
the part. of the instrument, or of the observer, we can
only cali either good, when positive errors are as likely

ON ERRORS OF OBSERVATION. 147

as negative. The average of a large number of observ-~
ations will in such a case be extremely near the truth,
and provided this condition can be fulfilled, the absolute
amount of the tendency to error is comparatively un-
important. Let two observers, A and B, have instru-
ments the average error of the first of which is double
of that of the second. A given number of observations
made by A is not as likely to be within a given amount
of the truth in the average as the same number made
by B. But the former may more than make the
difference, by taking a larger number of observations.
The rule is, that the square roots of the numbers of
observations must be in proportion to the average errors
of the instruments. That is, if A’s instrument have an
average error double of that of B, he must make four
times B’s number of observations before he can place
the same reliance upon his own observations which he
ought to do upon those of B. And the same is true
if for average error we read mean risk, or probable
error. Butif the weights of the observations be known,
the numbers of observations (and not their square roots ),
must be inversely as the weights.

When, however, there is a fixed error in the instru-
ment, independently of casual errors, or of such as are
as likely to be positive as negative, there are two
modes of proceeding. The average of the observations
will now be too great or too small, according as the fixed
error is positive or negative.

1. If the truth can be found by any other means,
in any one instance, a large number of observations,
such as would be made if the truth in that one in-
stance were the object of inquiry, will serve to detect
the fixed error, with a high degree of probability that
the result shall be correct. If a result should be 29
and the average of 100 observations give 28, then it
must be presumed that instead of errors + # and — a
being equally likely —1-+w# and —1—w are equally
likely, or there is a fixed error of —1. If A be the
true result, and if P be the fixed error, then A + P is

L 2

14& ESSAY ON PROBABILITIES,

the result which the instrument would give, in the long
run. In p. 137. is shewn the method of determining
the chance that in s observations, the average should lie
within e of the final average, that is, within eof A + P.
This ascertains the chance that the final average just
mentioned lies between A + P+e andA+P—e, Let
R be the result shown by the instrument, the truth A
being otherwise known. Then if R lies between
A+P—eandA+P+e, it follows that P lies between
R—A-+eand R—A —e, and the chance of the first
is that of the second.

Exampie. The truth being known to be 30, and
the average of 20 observations giving 31, what is the
chance that there is in the instrument a fixed error
lying between 1 + 1 and 1 — }, or between 3 and 9.

The weight of the observations must first be found,
which is done by summing the squares of the errors,
taking the average given by the instrument as true,
precisely in the manner star in p.137. Suppose this
weight to be 10; then e or 4, multiplied by the square
root of 10, or 3: 162, is 79, to which in table I., the
value of H is‘74. It is therefore about 3 to 1 that
the instrument, in the long run, would give a result
between 31 + } and 31—1; that is, that there is a
fixed error in the instrument lying between 1 + 4 and
Lior de
This first method, then, of ascertaining the fixed
error of a set of observations, supposes that there are
cases in which the result is known beforehand, so that
the instrument may first be read by the aid of phe-
nomena, instead of phenomena by the instrument. The
first observation is that of the error of the latter, found
by comparing its indications with the known truth ;
the second, the observation of unknown phenomena,
follows: accurate results being obtained, not by altering
the instrument, but by applying the correction to the
observations which the preceding class of observations
has rendered necessary.

A reader unused to astronomical works, on opening

ON ERRORS OF OBSERVATION. 149

a book on the practical part of the science, might imagine
that no part of the subject pretended even to ordinary
accuracy. Nothing appears to be done which is unaf.
fected by serious error; and it seems as if a little more
attention to the fabrication of instruments would render
nine tenths of what has been written altogether useless.
This appearance is the victory of the head over the
hands ; the means of detecting the errors of instru-
ments are much more powerful than those of correcting
them. It is also the victory of astronomy over the
other physical sciences, on our knowledge of which the
manufacture of utensils depends: we know more of
the laws which regulate the changes of the heavens
than we do of those on which the stability and fluctu-
ation of instruments depend. Nor does the semblance
above mentioned entirely spring from unavoidable
error: for it is frequently the most convenient plan to
allow an error to subsist which might be corrected at
once, but which may be more easily corrected at
another stage of the process. It is also sometimes
useful to allow an error to remain of a larger magnitude
than is physically necessary, if by that means another
risk of error may be avoided. For instance, if re-
quisite correction be either an addition ar a subtraction,
sometimes one and sometimes the other, the most
practised calculator will be very liable to confound the
two. This may be remedied by allowing to the instru.
ment a fixed error, either additive or subtractive, of
such a magnitude that casual fluctuations will never
alter its name. The correction, therefore, will always
require the same process, and the risk of error arising
from taking the wrong method will be avoided.

2, The next plan of eliminating the fixed errors
of an instrument is by giving it such a construction
that an observation can be made in two different ways,
in which the fixed error must necessarily have dif.
ferent signs, and must be of the same amount in both
cases. This is in reality a method of making the
positive and negative errors of the same amount in

L 3

150 ESSAY ON PROBABILITIES,

the long run. Suppose, for example, that (as in the
transit-instrument) the correctness of the observations
depends (partly ) upon the line of sight of a telescope being
always exactly perpendicular to the axis upon which
the telescope turns. Such exact perpendicularity is
a mathematical fiction, which was never yet realised :
the telescope will incline more or less to the right
or to the left. But if the telescope be fixed to its
axis, and if the axis itself rest on pivots, from which
it can be taken off and the position of the instrument
reversed, it is obvious that such a reversal of the ends
of the axis will alter the error of the instrument,
throwing the line of sight as much to the left as it
was before to the right, or vice versd. The average of
a large number of observations will now present no
signs of fixed error, arising from this cause at least :
provided that the numbers of observations made in the
two different positions be equal. The chance of the
average of s observations in each position lying within
a given degree of nearness to the truth is precisely that
of twice s observations made with no fixed error, and
the same tendency to positive and negative casualties
as before. When the result has been obtained by
combination of the different sets, the fixed error of
the instrument may be ascertained by comparing the
combined average with the separate averages of each
set. If the observations be numerous, and the reversal
of the method of observing introduce no new errors,
then the combined average will be an arithmetical
mean (or nearly so) between the other averages, and
the difference between the former and either of the
latter will be the fixed error required.

Independently of one or other of these two methods,
the only result directly furnished by the observations
(except the average affected by the fixed error) is
their weight, which is obtained precisely as in p. 137.
It will be seen that either of the preceding methods
introduces entirely new elements ; in the first we have
previous truths for comparison with subsequent ob-

5

ON ERRORS OF OBSERVATION. 151

servations ; in the second, an adaptation of the instru.
ment which reproduces an equal likelihood of positive
and negative error, by making the fixed error itself as
often positive as negative. What we have called a
fixed error is in fact a part of the phenomenon, styled
an error because it is not a part of the result we wish
to observe. The errors which a simple application
of our theory removes are those of which no account
_ whatever can be given, and of which nothing can be
previously known. Such is not the case when positive
error is more likely than negative, or vice versd: for
this very circumstance is itself a phenomenon, which
must arise from some unvarying cause.

Having stated that it is indifferent, in a mathematical
point of view, whether the law of the facility of error
above explained be true or not, because any law what-
soever which falls within the widest permission of
common sense leads to the same results as above ex-
plained, when the number of observations is consider-
able—I will now point out, in some simple cases,
how different laws of error are to be reduced to the
preceding.

1. Let all errors, positive and negative, between + E
and — E be equally probable, and all others impossible.
Treat large numbers of such observations as in pages
187. and 144., on the supposition that the weight is 3
divided by twice the square of E.

2. Let the probability of error decrease uniformly
as the magnitude increases, the greatest possible errors
being + E and —E; which implies that the chance
of an error lying between — # and + 2 is the product
of w and the remainder of 2 E divided by the square of
K. For instance, if E be 10, and this law of facility
prevail, the chance of an error lying between — 2 and
+ 2,is the product of 2 and 18 divided by 10 times
10, or =°5- In this case a large number of observ-
ations must be treated as if the weight of each observ-
ation were 3 divided by the square of E.

3. If the weight of the observations be considered

L 4

152 ESSAY ON PROBABILITIES.

to be 5 divided by twice E’, the preceding methods
will show the chances of a large number of observ-
ations, upon a supposition intermediate between the
two last, and coinciding nearly with the first when the
errors are small, and with the second when they are
considerable.

I now proceed to the method of combining the
results of observation, and deducing the mean risks of
error, Suppose, for example, that A and B are two
results of a large number of observations, of which the
product is required. Nothing can be more erroneous
than to suppose that the mean risk of this product
will be the product of the mean risks of its factors,
By the mean risk here is meant the same thing as
before: imagine the product of A x B formed in every
possible way from the single results of the several
observations. Each error will be as likely to be positive
as negative, if the errors of the original observations
be the same. Take the average of all the several
errors, neglecting their signs, and one half of this
average will be (in the long run) what is called
the mean risk, explained in the same manner as in
p. 135.

The risk of the result will be modified by the
manner in which the operation makes one quantity
affect the error of the other. Suppose, for example,
that observation gives A = 100, and B= 150. If
these were certainly true, the required result would be
accurately 100 x 150, or 15,000. But if 100 be
wrong to any amount, the product will be wrong by
150 times that amount, on that account only: while,
if B be wrong, its error will be multiplied 100 times
in the result ; besides which, there will be an additional
error, the product of the two errors of A and B. On
the other hand, if observation give A too large and B
too small, the opposite errors may either compensate
each other exactly, and give the product precisely what
it ought to be, or may make some approach towards this
compensation. The product then may be rendered

eae ete

RA ees het OEY ae Ae pee eee

ON ERRORS OF OBSERVATION. 153

much more erroneous than the observations, or much
less so ; both of which possible cases are considered, in
all their extent, in the investigations which give the
following results. In all of them, except the first, the
mean risks are supposed to be small.

1. To find the mean risk of the sum or difference
of any number of quantities determined by observation,
add together the squares of all their mean risks,
and extract the square root of the result. Thus, if
the mean risks of two quantities be 3 and 4, that
of their sum or difference is the square root of 16 + 4,
that is, 5. If the mean risks be all equal, the rule
may be simplified into that of multiplying the mean
risk of one by the square root of the number of
quantities. Thus the mean risk of the sum of 100
observed quantities of equal risk is 10 times that of
one of them.

Exampue. Given the mean risks of A, B, and C,
namely, 1, 2, and 3, required that of 1OA+9 B-—4C,
Here every error which can happen in A is made
tenfold in 10 A, and the mean risk of 10 Ais 10x 1
or 10. Similarly the mean risks of 9 B and 4, are
9x2 and 4x3 or18 and 12. The squares of 10,
18, and 12, added together, give 568, the square root
of which is 23°8, the mean risk required.

It may seem strange at first sight that, ceteris paribus,
the mean risk of a sum and difference should be the
same. But a little consideration will show that, posi-
tive and negative errors being equally likely, the errors
of a difference may be as large as those of a sum:
and that no combination of errors can affect a sum,
without an equal probability of another equally pro-
bable combination affecting the difference in the same
way. |

2. To find the mean risk of the product of any
number of quantities A, B, C, &c. Take the fraction
which each mean risk is of its quantity: add the
squares of these fractions, and multiply the square
root of the result by the product itself. This rule is

154 ESSAY ON PROBABILITIES.

only to be trusted when the mean risks are small.
Let A= 100, B=150, and let the mean risks of A
and B be 1 and 2. Then 1 is ‘01 of 100, and 2 is
°0133 of 150. The squares of *01 and ‘0133 are ‘0001
and 00017689, the sum of which is -00027689, the
square root of which is ‘0167. This multiplied by
100 x 150, or 15,000, gives 250°5, which is the
mean risk of the product 15,000.

4. To find the mean risk of a fraction, or of the
quotient of a division, multiply each term (numerator

and denominator, or dividend and divisor) by the mean’

risk of the other, add the squares of these products and
extract the square root of the sum: divide this by the
square of the denominator or divisor; the result is
the mean risk required. But if the fraction be very
small, it is sufficient to divide the mean risk of the
numerator by the denominator ; while if the fraction
be very great, it is sufficient to multiply the fraction by
the risk of the denominator, and to divide the result by
the denominator.

The preceding will serve as specimens of the manner
in which complicated results of operation can have
those probabilities investigated which depend upon the
probabilities of error in their constituent parts. It
would be impossible to lay before a reader unacquainted
with the differential calculus, any such digest of rules
as would enable him to treat all cases with facility.
Any one of the mean risks obtained above will serve to
determine, as in p. 139., the weight of the result, from
which its law of error may be investigated, as in
p. 143.

It appears that the chances of error may be con-
siderably multiplied in the course of the operations to
which the results of observation are subjected. It
must, therefore, be the object of an inquirer not only
to make good observations, but also to select such
methods of observing, and such methods of treating
the observations (the latter generally depending upon
the former), as will render the final error the least

a
5 eS
=o

ON ERRORS OF OBSERVATION. 155

possible. The considerations necessary for this pur-
pose form a great part of the application of mathe-
matics to the sciences of observation: in which it
frequently happens that good methods of observing
are rendered useless by the multiplication of error which
the methods consequent upon them involve: and con-
versely, that formule good in other respects, are in-
admissible from the tendency to error in the observations
which they require. And it has happened before now
that mistakes of serious amount have arisen from the
use of mathematical methods in which the errors of the
observations are much multiplied.

I could hardly close such a chapter as the present
without some mention of the celebrated method of least
squares, on which the astronomy of the last thirty
years has depended for much of the increase of ac-
curacy which has been its characteristic. But as
any development of this very interesting subject is
impracticable without recourse to mathematical symbols
and reasoning, I content myself with a description
of one particular case, which is of very frequent
occurrence.

Suppose a number of results to be obtained by observ-
ation, from which a consequence is to be drawn by mathe-
matical reasoning. If the observations were all correct,
the consequence deduced from any one would be the
same as that from any other; but owing to the
errors of the observations, such agreement is of course
unattainable. It is, therefore, a question what method
of combining the several results should be adopted :
and mathematical analysis shows that the object is
attained by choosing such an intermediate result as
shall make the sum of the squares of the errors the
least possible. It might seem as if, positive and nega-
tive errors being equally probable, the average of results
is the most probable truth ; and this is the case when
the observation is itself made directly upon the result
which is required, or when there is only one datum
into which the uncertainty of observation is intro-

156 ESSAY ON PROBABILITIES.

duced. But even in such a case we have no right to
say that the average is preferred to the result of the
method of least squares; for the former is then a par-
ticular case of the latter. Let three observations give
9, 11, and 16, the average of which is 12. The
errors, taking this average as the truth, are 3, 1 and 4,
the sum of the squares of which is 26. This is the least
possible sum of the squares. To try this, assume 11 as
the most probable truth: the errors are then 2, 0, and
5, the sum of the squares of which is 29: assume 13,
and the errors are 4, 2, and 3, the sum of the squares
of which is 29. To avoid introducing fractions, I
have only assumed whole numbers, but if I take 121
or 11°9, I find the sum of the squares of the errors to
be in the first case 26°03, and in the second 26:03.
So that when one result only is in question, a direction
to take the average is equivalent to a direction to
make the sum of the squares of the errors the least
possible.

Let us now suppose two results of observation, say
that we wish to know the fraction which A is of B,
where both A and B are subject to errors, the positive
and negative being equally likely. Suppose, for ex-
ample, that we ask for the proportion of the population
of a country which is buried in a year. Statistical
returns will furnish the population of each year, and
the burials, both subject to errors. ‘There are now
obviously two ways of taking an average; I may
either divide the average burials by the average popu-
lation, or find the proportion which the first is of the
second in each year, and take the average of the pro-
portions. .One not versed in mathematics would sup-
pose that these must give the same results, but any
simple instance will show the contrary. The average
dividend and the average divisor do not give the average
quotient. For instance, let dividends be 12, 13, and
17, and divisors 20, 22, and 30. The fractions 43,
13, and 47,are°6, 591, and °567, the average of which
is ‘586. The average dividend is 14,the average

ON ERRORS OF OBSERVATION. 157

denominator 24 and 4+ is not °586, but °583. The
results nearly coincide, but so do the data, for which
reason the most probable result may be very nearly
found, or of two results which differ very little, one
may be much more probable than another. When
observations give magnitudes so nearly coinciding as 6
°591 and -567, it is worth while to examine the relative
probabilities of methods which give results so nearly
equivalent as *586 and °583. Which of the two
preceding methods is most entitled to confidence ?
Analysis points out that this question is useless, because
there is a third method which is more safe than any
other. The method of least squares in the case before
us leads to the following rule;—-When both the
numerator and denominator of a fraction are to be
determined by observation, and various corresponding
observations of both are made, multiply each numerator
and denominator by the denominator, and divide the
sum of the numerators so formed by the sum of the
denominators. ‘Thus in the preceding instance, it is

12 420 +19 x 24.417 x50. 1086
20 x 20+22 x 22+30x30 1784

or °581

which is more probable than either °586 or °583.

If the mean risks of all the observations be the same,
the mean risk of the preceding result is found by adding
1 to the square of the result obtained (-581) dividing
by the denominator which produced it (17 aa extract-
ing the square root of the quotient, and multiplying
the mean risk of each observation by this square.
Thus 581 x ‘581 is °337561, which divided by 1784
gives ‘000189, the square root of which is -014. The ©
mean risk of the result, therefore, is less than one
seventieth part of that of each of the observations.

The method of least squares is an extension of that
of taking an average, or rather it indicates the most
probable average in cases which, by reason of more
results of observation than one being involved, an in-
finite number of different averages exists. It is not

158 ESSAY ON PROBABILITIES.

yet introduced into the affairs of common life, though
many cases occur in which it might be made useful.
But many things which are only demonstrable by the
higher branches of mathematics are looked upon as
useless by those who do not understand them; nor
is this result of ignorance only to be looked for
among the uneducated. While the Reform Bill was
in its progress through the House of Commons, a
method was suggested by a man of science, with whom
the government advised upon the subject, for esti-
mating the relative importance of boroughs by con.
sidering their population and contributions to the
revenue combinedly. This method, to the efficiency of
which most of those who examined it gave strong testi-
mony, was ridiculed by some members of the house,
partly because it involved decimal fractions, and partly
because another and a more simple (but palpably
wrong) method gave, in that particular case, nearly the
same results. When legislators are neither able to see

that erroneous methods may sometimes lead to truth,:

being not therefore one bit the less erroneous, nor that
the truth of a result is the same, whether decimal or
common fractions be employed, it is little to be won-
dered at if useful applications of abstract reasoning
are looked upon with suspicion and introduced with
difficulty.

CHAPTER VIII.

ON THE APPLICATION OF PROBABILITIES TO LIFE CON-
TINGENCIES.

WHEN questions connected with life contingencies were
first considered, it was regarded as most deliberate
gambling to be in any way concerned in buying or
selling such articles as annuities, or any interests
depending upon them. Before we can well enter upon

Pe te

seni

ON LIFE CONTINGENCIES. 159

the question of the truth or falsehood of the preceding
notion, it will be necessary to ask what laws the
duration of human life follows, and whether it fol-
low any laws at all? Take two separate hundreds
of persons, each aged twenty, is there any reason to
conclude that the united lives of all the first hundred
will make an amount of years nearly equal to that of
the second P

In order to try this point, I shall take another question,
yet more unfavourable to the result which I wish
to establish. In 100 persons all aged twenty, we know
‘that there is but a very slight chance that any given
one of them shall reach the age of eighty; and we
may consider it a certainty (or of an extremely high
probability), that none of them will see the age of a
hundred and twenty. We will consider it therefore as
given, that no one shall live to the last-mentioned age,
and we will even suppose that all ages of death between
20 and 120 are equally probable. This of course
very much increases our chance of fluctuation: but
even with this supposition it is not very great.

Let us suppose a lottery in which there are counters
marked with every possible number or fraction inter-
- mediate between 0 and E: so that the drawing may
have any mark whatsoever. If then we draw out 100
counters, the least possible amount of drawings will be QO,
the greatest 100 times E: and if all drawings be
- equally probable, we have no reason to suppose that
our amount will exceed 50 times E, which does not
equally apply in favour of its falling short of that
quantity. That we shall have exactly 50 times E, is
an event of which the chance is infinitely small: but
that the amount shall lie between limits which are
tolerably near 50 times E, is very probable.

Propuem. Let there be counters, in equal numbers,
with every possible mark between 0 and E. Whatis the
probability that the average of n drawings shall not differ
from the half of E, one way or the other, by more than &.

Ruue. Multiply & by the square root of six times

160 ESSAY ON PROBABILITIES.

m, and divide the product by E. Call the quotient t ; _

then the value of H (Table I.) is the probability re-
quired.

Exampre. In 600 drawings, each of which may
be any thing between O and 100, required the proba-
bility that the average of all the drawings shall lie
between 50+ 5 and 50— 5.

n= 600, E= 100, k = 5; the square root of 6 times
600 is 60, and 5 times 60 divided by 100 is 3. The
first table does not contain values of t higher than 2:
an event being almost certain, or of a very high pro-
bability when t is equal to 2. ‘Table I1., however,
furnishes us with an extension of Table I. ; the K oppo-
site to any value of t in that table being always nearly
the H which belongs to half that value of t. Conse-
quently, the H belonging to t = 3, is the K belonging
tot=6. But the second table only goes tot = 5;
in which case K is ‘999. It is then more than 999 to
1 that the average of the 600 drawings is within the
limits specified. If we take k= 1, in which case
t = °6, we find it is 3 to 2 that the average is con-
tained between 49 and 51.

If then there were 600 infants born, and if it were
the law of human life that any individual is as likely to
die at one age as another, for any age not exceeding 100
years, even then, and with so much more scope for
fluctuation than is actually found, it would be more than
999 to 1 against the average life of the 600 infants
exceeding 55, or falling short of 45 years ; and more
than 3 to 2 that the same average should fall between
49 and 51 years. If such be the case, it is obvious that
the chances of fluctuation are much diminished by the
superior chances of death happening at some. periods of
life rather than at others ; as well as by the smaller limits
of human life, which need not for any practical purpose
be supposed to extend as far as one hundred years.

To suppose that the duration of human life is regu-
lated by no laws, would be to make an assumption of a
most monstrous character, d priori, and most evidently

eB ate

ON LIFE CONTINGENCIES. 161

false. For it is a law, were it the only one, that no
individual shall attain the age of 200 years. So much
is known to all; but to those who consider the subject
more closely, by the aid of recorded facts, it may be
made as evident as the existence of a limit to human life,
that the casualties of mortality are distributed among
mankind in so uniform a manner, that the average
existence of a thousand infants will differ very littk
from that of another thousand born in the same country
and station of life. It is true that differences of race,
climate, manner of living, &c., &c., produce marked
effects upon the duration of life; which is no more
than might be expected: but it is equally true that the
notorious individual uncertainty of life cannot be dis-
covered in the results of observations made upon masses
of individuals. ;

There are various results of observation, which are
called tables of mortality, which differ only in the
methods of presenting the same sets of facts. Firstly,
we have what may be called tables of the numbers living.
_ These show, for a given number born, how many attain
each year of age. Thus, in the Carlisle table, opposite
to O and 50, we find 10,000 and 4397, indicating that,
according to observations made at Carlisle, the propor-
tion of those born to those who saw their fiftieth birth-
day, was that of 10,000 to 4397. Again, opposite to
60, we find 3643, meaning, that of 4397 persons aged
50, 3643 attain the age of 60. Secondly, we have tables ©
of yearly decrements, in which the same number of per-
sons are supposed to be alive at every age, and the pro-
portion who die in the next year is set down in the table.
Thus in the government annuity tables, opposite to 50
and 60, we find 161 and 315, meaning that, according
to the observations from which these tables were con-
structed, of 10,000 persons aged 50, 161 died before
completing the next year of life; and of 10,000 persons
aged 60, 315 died before attaining the age of 61.
Thirdly, we have tables of mean duration of life (com-
monly called expectation of life), which show the average

M

162 ESSAY ON PROBABILITIES.

number of years enjoyed by individuals of every age.
This, in another variety of the Carlisle tables, opposite
to 50 and 60, we find 21°11 and 14°34; meaning that,
according to these tables, persons aged fifty live, one
with another, 21:11 years more, and persons. aged 60,
14°34 years more.

Until observations of human mortality become more
extensive and correct, I prefer the tables of mean
duration to all others. The events of single years are
subject to considerable error, and generally present
such varieties of fluctuation, that it has become usual to
take some arbitrary and purely hypothetical mode of
introducing regularity. This practice cannot be too
strongly condemned, since the tables thereby lose some
of their value as representations of physical facts,
without any advantage ultimately gained. For if by
using the raw result of experiments, tables of annuities
were rendered unequal and irregular, it would be as
easy, and much more safe, to apply the arbitrary method
of correction to the money results themselves, than to
introduce it at a previous stage of the process. It is
not, however, a matter of much consequence as to the
annuities, &c., deduced from the tables: and as yet, the
rudeness of the original observations renders the effect
of any such alteration not so great as the probable
errors of the observations themselves.

The mean duration of life is approximately calculated
as follows. Suppose (taking an instance from the Car-
lisle tables) that '75 persons are alive at the age of 92,
of whom are left at the successive birthdays, 54, 40,
30, 23, 18, 14, 11, 9, 7, 5, 3,1,0. Consequently, in
their 93d year, 54 persons enjoy a complete year of life,
and 21 die, whom we may suppose, one with another,
to live through half the year, and 54 years and 21 half
years make 644 years, which is the total life of 75 per-
sons for that year. Proceeding in this way, we find
that there are,

in the 93d year 54 + } of 21 years.
94th 40 + dof 14

ON LIFE CONTINGENCIES. 163

95th 30 + tof 10
96th 28+i0f 7
97th 18 + i of
98th 14+i0f 4
99th ll +iof 8
100th 9+t0f 2
101st 7+;0f 2
102d 5 +iof 2
103d $3+3,0f 2
104th l+4of 2
105th O+40f 1

Total, 215 + 3 of 75

Hence 75 individuals, aged 92, enjoy 215 + 4 of 75
years, and each has, one with another, the 75th part of
this, or 3°37 years.

Rutz. To find the mean duration of life from a
table of the living at every age out of a given number
born, add together the numbers in the table for all the
ages above the given age, divide by the number at the
given age, and add half a year to the result.

The preceding rule is mathematically incorrect, being
only an approximation to the truth, even supposing the
tables perfectly correct. The error of computation
may be found, nearly, as follows. Divide the number
who die in the year next following the given age by
twelve times the number in the table at that age, and
diminish the result of the preceding rule by the quotient.
Thus, in the instance before us, 21 divided by 12 times
75 is ‘02, so that 3:35is nearer the truth. ‘This error,
however, is immaterial for practical purposes.

A more important question is that of the degree of
confidence which may be placed in tables of mean
duration, the errors of observation. being supposed to be
as likely to be positive as negative. In order to
estimate this, we must compute the mean square of the
duration of life; that is, multiplying the time which
each individual lives by itself, we must add the results
together and divide by the whole number of individuals.
To make a rough approximation to this in the case
before us, remember that

164 ESSAY ON PROBABILITIES.

21 individuals live } a year

5X 4 = 7 giving 2

14 3 hee tae 8 4
5 ‘ko es 250

10 —_ oo 7. ee
ee ¢ RS. | Agree Lf

fj ten Hf hx ear: a
ee vam 405

5 — 3 — | ee cates Geen rc
ene AO re tS eee

4 h § 2* 9 4 ar 3
3 Ms ee eee. ee
apne! 2 of 3° 8 4

9 ee iu. By ites ie
2 Yaa eae 4

2 ee wZ— Wy 17—289 . 578
2 .: 22 4 4

2 “Teor ter 4

2 ae 2 2. ‘Aya ok
2 Your eer’ 4

9 ca 23 23 y 23529 __ 1058
2 Boas epee q

l oes 3 8. 8,8. _ om
2 ca oe 4

Average square 21° 6451

75 ( Average sq 5) ri

Rutr. From the mean square of the duration of
life at any age, subtract the square of the mean duration
at that age: divide the difference by the number of
lives of the given age from which the table was made,
and extract the square root of the quotient. Take
four tenths (more correctly °39894), of this square
root, which gives the mean risk of error, and ‘67 of
the square root gives the probable error.

Suppose that in the case before us, the number of
lives aged 92 was 40 *, from which the preceding table
was made. We have then,

Mean square of durations 21°5
Square of 3:37, the mean duration 11°36

40)10°14
"254 V°954= *504
*504 x 67 = ‘33 of a year, the probable error.

The same process may be applied to any other case,
and the result of the whole is, that observation of a
number of lives which is not very great, will be suffi-

* This is nearly the number of lives at that age among those from
which the Carlisle table was formed, but the arbitrary help introduced
from other tables at the older ages, on account of presumed insufficiency of
data, makes the result of this example of no greater value than a numee
ricai instance arbitrarily chosen.

Ne eT a ae

ON LIFE CONTINGENCIES. 165

cient to give the mean duration of life with considerable
approach to exactness. This is confirmed by the
results of various tables, from which it appears that
when the individuals composing an observation are of
the same country, and under the same general circum-
stances, the results of such tables come very near to each
other.

The reader who desires to know the history of tables,
of mortality should consult the articles Morrauiry and
ANNUITIEs in the new edition of the Encyclopedia
Britannica, both from the accurate pen of Mr. Milne, the
author of the Carlisle tables. I cannot, in this work,
pretend to give more than a slight summary of results
connected with life contingencies, such as niay guide the
reader who understands the main points of the theory
of probabilities to safe conclusions.

From some tables made from observations at Breslau,
De Moivre concluded that the following hypothesis,
namely, that of 86 persons born one dies every year
till all are extinct, would very nearly represent the
mortality of the greater part of life, and that its errors
would nearly compensate one another in the calculation
of annuities. The Northampton tables of Dr. Price,
which have been used by most of the insurance offices,
very nearly represent this hypothesis at all the middle
ages. But both give much too large a mortality for
the circumstances of the last half century, as is proved
by all the tables which have been lately constructed.
The greater part of the difference, I have no doubt, is
due to the real improvement of life which has taken
place, from the introduction of vaccination, more tem-
perate habits of life *, better medicai assistance, and
greater cleanliness in towns. We may now state, as a
much nearer approximation to the mortality of the

* I must be understood, here, as speaking particularly of the middle
classes, in English towns and cities, Most of the tables have a majority of
this class, and there is not any very precise information on the mortality
of the labouring classes, or in the inhabitants of the country as distin-
guished from those in towns. With regard to the point on which this
note is written, all old persons remember the time when what we should
now call hard drinking was almost universal.

mu 3

166 ESSAY ON PROBABILITIES.

middle classes, that from the age of 15 to that of 65,
the average may be represented as follows: —of 100
persons aged 15, one dies every year till the age of 65.
But the mean duration of life will serve to give a more
precise idea, and a simple rule may be given, which
will, for rough purposes, represent the Carlisle table
between the ages of 10 and 60. Of persons aged 10
years, the average remaining life is 49 years, with a
diminution of 7 years for every 10 years elapsed ; thus
of persons aged 20 years, the average remaining life is
49-7 or 42 years ; at 30 years of age, 35 years. The
following list of tables will be followed by some notice
of each.

% E § 3) o rf £ g,
cs) WM « = o = o n
=| fe] 8] 3/138) 2 |22 | gels
pee e188 oe | ee eel
inl We i Bee! el
a
O | 43 Bhs fe fe") 387 Le ie Le ce ee O
§ | 40°5 | 408 | - - | 513 |= - | 48°9 | 54:2 5
10 | 38 39°8 |= = | 48°8 | 48°3 | 45°6 | 51*1 | 10
15 | 35°5 | 36°5 - | 45°0 | 45°0 | 41°8 | 47°2 | 15

70 |8 8°6 78.) 92 |. 8:7.) 92 1 11.0 199
75 15:5 | 6:5 621-70 | 66.[ °71.| 83 te
80 | 3 4°8 50 | 55 | 48 | 4:9 | 6:5 | 80
85 |0°5 | 394 40] 411] 8 3:1 | 4:8 | 85
90 |- - | 2°4 29 | 33 | 261]: 2:0 | 2:84 90
95 |= - | 08 14] 395] 11 172 | 1:6 | 95
100%}~ = |- si. D3 Vie. cmt inky Jen ew 1100:

ON LIFE CONTINGENCIES. 167

1. De Moivre’s hypothesis was suggested by Halley’s
Breslau tables, made from observations of the mortality
of that town in the years 1687—1691. It confessedly
errs considerably at the beginning and end of life.

2. The Northampton tables were constructed by
Dr. Price from the mortality of that town, in the years
1741—1780, the numbers of male and female deaths
being very nearly equal. These tables were, and are.
almost universally used by the assurance offices, ana
are those by which legacy duties are estimated in the
act of parliament, 36 Geo. III. cap. 52.

3. The Amicable Society's table was formed some
years ago by Mr. Finlaison, at the suggestion of that
gentleman and myself to the directors, and as a means
of furnishing information upon points * as to which
they had consulted us. The Amicable society was
founded in 1705, and the table is formed from the
experience of more than half the subsequent period
ending in 1831.

4. The Carlisle table, formed by Mr. Milne from
the observations of Dr. Heysham upon the mortality of
that town, in the years 17791787. They are to be
considered the best existing tables of healthy life which
have been constructed in England. The relative pro-
portions of the sexes are 9 females to 8 males.

5. The Equitable table (published by the Equitable
society in 1834) gives the results of the experience of
that society from 1762 to 1829. The total number or
deaths recorded is upwards of 5000.

6. The Government tables (male and female life
separately). These tables were constructed by Mr. Fin-
laison, actuary of the national debt office, from various
tontines, &c., of which the records are in the possession
of the government. Each table contains about 5000

* In mentioning this subject, I may be allowed to state my full approval
of the plan subsequently adopted by the society, and my copviction ‘that
the errors of their ancient system haye entirely disappeared.

The mean durations above given were computed by myself, from the
tables of decrements circulated by the directors among the members.

uw 4

168 ESSAY ON PROBABILITIES.

deaths. These are the tables on which the commis-
sioners for the reduction of the national debt grant life
annuities in lieu of stock.

I will now add some deductions made by myself
from the tables contained in the Recherches sur la
Reproduction, &c., &c. Brussels, 1832, by M. Que-
telet and Smits; republished in the treatise Sur l’ Homme,
&c. of the former. They are founded upon the
statistical returns of the whole of Belgium, made in
three successive years, and distinguish not only the
sexes but the residences of the parties, whether in
towns or in the country. The middle table is the
general average of the whole country, whether male or
female, in town or country.

Age. Towns. Both. Country. Age.

Males. |Females.} Both. Males. |Females.

0 29°2 | 3$3°3 32°2 32°0 | 32°9 O

5 45°0 | 47°1 45°7 46°1 | 44:8 5
10 42°9 | 45°0 43°9 44°4 | 42°9 10
15 $90 | 41.3 | 40:5 41°2 | 40:0 15
20 35°4 | 38°0 37.3 33‘I | S70 20
25 33°1 | 350 34°7 SFT 1 Sea 25
30 30°4 | 32:1 32°0 33°0 | 31°5 30
35 27°5 | 29°2 28°9 29-7. | 287 35

go | 48 | 51 5-0 50 | 51 | 80
85 | 37 | 40 3.8 38 | 3:83 | 85
90 | 2:9 | 3-0 3:1 3:1 | 3:21 90
95 1-3 | 2-0 21 22 | 1:9 | 95

100 | 00 | oO 1°3 0-5 | 05 | 100

ON LIFE CONTINGENCIES. 169

Most tables in which the sexes are distinguished
unite in presenting this result, that female life is ma-
terially better than male life. But this fact is much
more distinctly apparent in towns than in the country,
and in the Belgian tables the phenomenon is reversed,
so that while female life is decidedly better than male
life in the towns, it is not so good in the country.
Mr. Milne has remarked that in Stockholm the dif-
ference between male and female mortality was three
times as large a per centage of the whole as it was in all
Sweden. The probable reason for this discordance
is the different employment of women in town and
country; all the tables yet constructed which distin-
guish the sexes, and include rural life, having been
made from a great preponderance of the working
classes. The only tables which separate the sexes, and
which are formed from the middle classes, are those of
Mr. Finlaison ; and here the difference is greatest of
all.

This consideration is very material in comparing the
tables which I have given. If a table of male life
should fall short of one of female life, all other cir-
cumstances remaining the same, it is no more than we
might expect ; while at the same time the true pro-
portions of male and female life, as well as the manner
in which they depend on local or other circumstances,
are very imperfectly known. But if a table of male life
only should present the same results as one of mixed
lives, we are then sure that the former represents a
longer duration of existence. For instance, the table
of the Equitable insurance office, which is almost entirely
composed of males, is almost identical with the Carlisle
table in which there are more females than males.
This shows that the select male lives of the office are
much better than the male lives of the Carlisle table:
ebut that the male lives of the office, constantly recruited
as they have been with selected lives of all ages, are
no better than the mixed lives of the Carlisle table.
Similarly, the males of the Amicable table are very

170 ESSAY ON PROBABILITIES.

much better than those of the Northampton table.
The male and female lives of the latter are nearly equal
in number ; the former is almost entirely founded upon
male lives: while the former, with its male lives only,
gives a longer duration of life than the latter. For
old lives, however, the Northampton table gives a some-
what longer duration than the Amicable. This is only
one fact out of many which show that the Northampton
table, while it gives much too great a mortality to the
younger class of lives, errs in the other extreme as to
the older. Of thirty-five tables, made in different
countries and at different times, and including ail of
any celebrity which had appeared before 1830, I find
that the Northampton table is the eighth from the
lowest at the ages of 10—25, and the tenth from
the highest at the age of 65. The same may be said
of De Moivre’s hypothesis, which the Northampton
table closely follows.

The Northampton and Amicable tables are decidedly
older as to the period at which their members lived,
than the Carlisle and Equitable. Life is shorter in the
former pair than in the latter, while both of the former
agree in presenting the older lives comparatively better
than the younger ones, as compared with the latter
pair. By the Northampton table, the duration at 65
years is about a third part of that at 25; while the
same proportion is decidedly less in the same ages of
the Carlisle table: a similar result appears in the
Amicable and Equitable tables. I remember remarking
the same phenomenon in the results of a comparison of
the lives of naval officers. There can be little doubt that
the reason is as follows: in circumstances which create
a large mortality at the younger ages, all the feeble
constitutions are prevented from attaining old age, so
that the lives which really arrive at advanced years are
the remains of the very best lives. I saw the ocur-
rence of the same disproportion in the lives of officers
of the Anglo-Indian army; in which, however, it was
probably increased by the residence of many of the

wip aes
ener

ON LIFE CONTINGENCIES. 171

officers in question in England during the latter years
of their lives.

The Equitable society has the character of having
been much more careful in the selection of its lives than
was the Amicable society during the earlier part of its
existence. This, together with the gradual improve.
ment of human life, serves to explain the very great
difference between the results of their experience. The
latter years of the Amicable society do not exihibit any
very decided difference of the sort.

The state in which we stand with respect to tables
of human life is singular, considering the enormous
amounts which daily pass from hand to hand in the
purchase of life interests. I may have occasion to
speak more at length on this subject in the sequel:
in the mean time let the reader observe the difference
between the various tables, and remember that each has
its votaries. If the late Mr. Morgan (whose name
stands very high as an authority on such matters) had
been requested to state the value of an annuity on the life
of a female aged 40, and the same for a male of the
same age, he would have replied that there was no
material difference between male and female life, and
that both belong to a class whose average existence is
23 years; and he would accordingly have used the
Northampton tables of annuities. At the national debt
Office, it would have been answered by Mr. Finlaison
that the male and female life are two very distinct
cases, and that the two different classes to which they
belong have severally the average lives of 27 and
31 years. That such differences should exist, is a
proof of insufficiency of information upon the sub-
ject: a want which nothing but the government can
supply, but which no government ever will attempt to
supply until increasing knowledge among the commu-
nity at large creates an influential body of remonstrants.

Having given a table of mean durations, it is easy to
find the proportion who die in one of the intermediate
periods, on the supposition that the deaths are equally

25 | 73|'76$50| 1841} 150475 | 955) 931

v2 ESSAY ON PROBABILITIES.

distributed through the period. This supposition is
not actually true, though for a long course of ages the
amount of mortality does not vary much from year to
year. The main feature of De Moivre’s hypothesis,
equal decrements, appears in some measure at the adult
and middle ages of life in all tables. Ido not, however,
know of any observations in which the numbers dying
at every age are large enough to produce much confidence
in the details of the tables of decrements, though the
fluctuations may compensate each other in the deter-
mination of the mean durations of life. Tables which
agree in the latter point may differ materially as to the
former. As an instance, I give the following com-
parison of the Carlisle and Equitable tables, which
agree more closely than any others in their mean
durations. The first column shows the common age of
10,000 persons, the second and third the number who
die in the following year in the Carlisle (C.) and
Equitable (E.) tables.

Pe Pee EAs) Co PE. PAST eC. EK. | A.}| € E

10 | 45| 724335|103| 92]60| 335] 315485 | 1753 | 2210

15 | 62| 75; 40/130} 110]65| 411] 428 | 90 | 2606 | 2686
20} 71 | 73}45|148|127]'70| 516] 639] 95 | 2333 | 5566

30 1101 | 81} 55 | 179 | 208 | 80 | 1217 ! 1329

According to the Carlisle table, of 10,000 persons
aged 30, 101 die before attaining the next birthday ;
while one fifth less die in the Equitable table. And
yet, one with another, the average lives of two sets of
10,000 do not differ by more than their 170th part.
I now compare the actual tables of decrements, writing
opposite to each age the survivors of 10,000 births who
attain that age. (A, age ; C, Carlisle; E, Equitable.)

de Xk. E. Anh ih E. Ai} t E.

0 | 10,000 35 | 5362 | 5292 | 70 | 2401 | 2310
5 6797 40 | 5075 | 5034 = 75 | 1675 | 1572
10 6460 | 6460 4 45 | 4727 | 4751 80| 953] 898
15 6300 | 6192 | 50 | 4397 | 4441 85 | 446; 354
20 6090 | 5956 4 55 | 4073 | 4069 90| 142 86
25 5879 | 5733 | 60 | 3643 | 3588 § 95 30 21
30 5642 |} 5524 § 65 | 3018 | 3002 4100 9

ON LIFE CONTINGENCIES. 173

In the Equitable table, 5000 persons are supposed,
each aged 10 years; this I have altered to 6460, to
make the two tables agree at their outset.

The successive quinquennial decrements of the Car-
lisle table from the age of 20 are 211, 237, 280, 287,
348, 330, 324, 430, &c. If these deaths be supposed
to take place at equal or nearly equal intervals during
the five years,— if, in fact, we may suppose each of the
individuals who die in a period to enjoy, one with
another, half that period of existence,— we may ascertain
the law of mortality from the table of mean durations
in the following manner.

Rute. To the mean duration at the end of the
period add the term elapsed and subtract the mean
duration at the beginning; divide by the smaller
duration increased by half the term, and the quotient
is the fraction which expresses the proportion dying
during the term. For example: the mean durations of
life at 25 and 30 in the Carlisle table are 37°9 and
34°3; and 34°3 + 5— 37: 93 is 1:4; which, divided by
34°3 + 2°5 or 36°8, gives 14 or =1, ; so that of 184
persons aged 25, 7 die ee attaining 30 years. In
the table, we have 43,5, while =4, is ehiut ATCA

If we were to take any table now existing on English
lives, and ask, (as in p. 92.), what is the probability
that a large number of lives, say 1000, should drop
nearly in the same manner as those from which the
table was formed, we should find the resulting chance
not strong enough to make it prudent to risk much
money in such contingencies. Nevertheless, the appli-
cation of this theory to pecuniary risks has always been
in a more forward state than the physical theory of
human life. The reasons will be explained when we
come to treat on the grounds of the confidence to which
a contingency office is entitled. In the mean while,
supposing a table to represent perfectly the average of
a large number of the lives of the class to which an in-
dividual belongs, I proceed to show the method of using
such a table

174 ESSAY ON PROBABILITIES.

Persons who are desirous of using tables of life on a
larger scale, are referred to the standard works of Messrs.
Morgan, Baily, and Milne, on life insurance. In the pre-
sent work I assume that it will be sufficient to be within
two years and a half of any age which may be named, and
I have given the several tables for intervals of five years.
The extremes which are used by actuaries generally
being contained in the Carlisle and Northampton tables,
and having given the former, I now add the latter.
The first column contains the age, the second the table
of decrements, the third the number out of 10,000 who
die in one year after completing the age in the first
column.

Age. Age. Age. |
O | 10,000 35 | $487 | 187 | 70 | 1056] 649
5 5356 40 | 3116 | 209 | 75 713 962

10 | 4864] 92 | 45 | 2784 | 2404 80 | 402] 1343
15 | 4648| 92 | 50 | 2449 | 284 1 85 | 159! 2904
20 | 4399| 140 | 55 | 2098 | 335 | 90 39 | 2609
25 | 4080| 158 } 6O | 1747 | 402 } 95 3 | 7500
30 | 3759] 171 | 65 | 1399 | 490

Supposing the tables perfectly accurate, the following
simple questions will show the nature of the first steps
which occur in their application. The Carlisle tables
are used throughout.

Question |. What is the chance that an individual
aged 35 will live to the age of 50? Of 5362 persons
aved 35, 4397 live to be 50; hence the chance in
question is #323 or °82. Answer, 41 to 9 for the
event.

Question 2. What is the chance that A aged 45
and B aged 50, shall both be alive in ten years? The
chance for A, by the last question, is 4°43, or 862
and that for B 3623 or +829; the product of these
(p. 43.), or *715, is the chance required. Again, the
chances of A and B dying during the ten years are
1— 862 and 1—-°829, or *138 and ‘171; whence,

ee ee 2

ON LIFE CONTINGENCIES. 175

The chance is
That both shall live "862 x °829
That A shall live and B die ‘862 x ‘171
That A shall die and B live ‘138 x -829
That both shall die 138 x °171

Question 3. What is the chance that A aged 25,
shall die between the ages of 60 and 65? Of 5879
persons aged 25, 3643—-3018 or 625, die between the
ages of 60 and 65; hence 3°25 is the chance re-
quired.

Questions of this kind are readily solved, the only
mpediment being the arithmetical operation. It fre-
quently happens, however, that the probability of one
individua! surviving another is required, which though
an even chance when the individuals are of the same
age, is a matter of considerable calculation, when one is
older than the other. Suppose, for example, that the
chance of A (aged 25) surviving B (aged 30) is required,
The survivorship, as it is called, meaning the period
during which A lives after the death of B, may begin
in any one year of A’s age. For each year the pro-
bability of the survivorship beginning in that year must
be calculated. To make this calculation for one in-
dividual year, say that in which A is between 49 and
50, two cases must be considered: either B may die
between 54 and 55, and A may attain 50 complete
years (of which the chance may be found as in the
preceding questions), or both may die in the same year
(that is A between 49 and 50, and B between 54 and
55), but B may die first. If the chance of both dying
in that year be, say,*012,it is sufficiently correct to con-
sider the half of this chance, or ‘006, as being that which
expresses the chance of A’s survivorship both beginning
and ending in that year: a supposition which is quite
correct only when the deaths of the year are equally
distributed through it. The result of this calculation
is arranged in tables, of which I here give a brief
abstract.

ESSAY ON PROBABILITIES,

ie He es ha 3 siete s
fo) 2 5 5 ~ = S 5
i615 “447 400 80 | 50 136 093
90110} +415 383 85 | 55 113 094
25115 | +420 331 90 | 60 097 113
30 | 20 | ‘423 375 95 | 65 | 037 147
35 125) °417 372 45°| 5 | 950 177
40 | 30]! °409 366 50 | 10 206 "146
45 |35 | 402 360 55 | 151] +201 "135
50 | 40 | ‘394 350 60 | 20 193 ‘119
55 | 45 | °385 329 65 | 25 172 110
60 | 50 *376 "315 70 | 30 148 *-097
65 | 55 *360 S23 75 | 35 124 ‘081
70 | 60 *339 322 80 | 40 102 ‘O75
95 1 (65.4. :°317 303 85 1451 -082 "059
80 | 70 *300 320 90 | 50 069 "052
85 | 75 | +292 "332 95 | 55 025 078

) 30 *32
a5 eg | ter | aon 2 | 81 108 [aa

jill ecaalh 60 | 10 144 091
25 | 5 ‘377 307 65°1 13 136 085
30 | 10] +344 283 T7091 901 198 7]
85115 | +345 279 75 \ 25 103 061
40 | 20] +343 270 89 | 30 082 056
45 | 25 331 263 85 | 35 064 "046
50 | 30 | +817 251 90 | 40 051 044
55 | 35 | +303 231 95 |45 |} -018 049
60 e ° ss

40 258 212 Ta ie cans 086

65 | 45 269 194
i 70 | 10] -087 054

70 | 50 246 177
534 V5 078 048

75 155 218 177
80 | 20 069 040

80 | 60 | +189 *190
; 85 | 25 053 034

85 |. 65 166 174
; . 90 | 30 041 033
90 | 70 157 191 95 | 35 014 039

95|75 | 072 300 :

“ae 314 a5 7S 1:54... 098 057
: 80110] -044 028

40|10]| -273 207
i 85 | 15 | +037 024

45;15| -271 202
pais 3 90 | 20 035 012

55 | 25 249 174
60; 30| -230 158 | 85 | 5 | 063 040
65-35 209 "143 90 | 10 022 ‘O17
70 |40| +186 126 | 95 | 15 007 "023
45 "160 104 95 | 5 021 ‘036

ON LIFE CONTINGENCIES. 177

The quantity found in this table is the probability of an
elder life surviving the younger. The difference of ages
differs in the various compartments of the table ; in the
first it is ten years, in the second twenty years, and so
on. The two results accompanying each pair of ages
are those of the Northampton and Carlisle tables. Thus,
according to the Northampton table, the chance of a
life of sixty surviving one of thirty is ‘23 ; that of the
younger surviving the elder is therefore 1 — +23 or °77.
According to the Carlisle table, the same chances are
°158 and °842.

Almost universally, the Northampton table gives a
greater chance of the elder life beating the younger than
the Carlisle. This is a consequence of that undue
degree of comparative goodness which the former table
gives to older lives, and to which I have already ad-
verted.

If De Moivre’s hypothesis were correct, it would be
sufficient to divide the mean duration of B’s life by
twice that of A, and the result would be the chance
which B has of surviving A, B being the elder of the
two lives. This process, applied to the Northampton
table, will give results very near the truth, when
neither of the lives is very young. The same rule
would give comparatively but a very rough guess at the
result of the Carlisle table. If, however, the chance be
calculated which the younger life possesses of dying in
the average term of the elder, the result will be an
approximation to the probability of the elder surviving
the younger, when neither of the lives is very young,
and when their ages are not nearly equal. Thus, the
mean duration of a life of 50 being 21 years, and the
chance of a life of 30 years surviving 21 years being
769, the chance of the same life not surviving 21 years
is ‘231 ; while in the table, the chance of a life of 50
surviving one of 30 is 251.

I shall, in the next chapter, consider the application
of the tables to pecuniary questions, and shall now

ie

178 ESSAY ON PROBABILITIES.

proceed to point out the connexion between a table of
mortality and one of population.

The whole number of persons inhabiting any country
is in continual state of increase from births and immi-
gration, and of decrease from deaths and emigration.
There are few countries in which immigration and
emigration produce any serious effect upon the popu-
lation, and, in times of very moderate quiet and pros-
perity, the births always exceed the deaths: so that,
generally speaking, the number of people alive in a
given country is yearly augmented by the excess of the
births over the deaths. If accurate registers of births
and deaths (with the ages at death) were kept for a
a century and a half, accompanied, if need were, by a
register of incomers and outgoers, with their ages, the
community would be in possession of a complete history
of its statistical changes, from which the law of mor-
tality might be deduced, and its fluctuations noted, if
any.

Again, if in any one year a complete census were made,
registering the age of every individual, and if the deaths
which took place in the 365 days next following the
day of the census were noted, the law of mortality
could be deduced. In such a case, the numbers of the
living at every age would be so large that the propor-
tion of deaths among them in a single year could be
safely depended on for pointing out, with great nearness,
the law which regulates the mortality of large masses of
people.

No such statistical means exist in this country,
partly from the defective manner in which the censuses
of population are made, partly from the circumstance of
the registries of births and deaths having been, almost
up to the time of writing this work, connected with
the religious ceremonies of the established church, which
has had the effect of excluding many dissenters from
registration. In the absence of all specific information,
recourse was had to the registers of burials, which are
usually accompanied by a statement of the age of the

ON LIFE CONTINGENCIES. 179

parties, though without any sufficient guarantee for the
accuracy of the information. The hypothesis upon which
alone registers of burials will give a correct law of mor-
tality, requires that one of two alternatives should exist :
either a permanent law of mortality, with a knowledge
of the population in every year, and of the number of
emigrants and immigrants, with their ages; or a
stationary population, with the same number of births
and of deaths in each year, and a permanent law of
mortality. This latter supposition is never exactly
true ; but, as many societies have made a near approach
to it, and as many tables have been constructed by its
means, it will be worth while to explain the conse.
quences of the supposition.

If the Carlisle law of mortality remained in unin-
terupted operation for a century, and if 10,000 infants
were born alive in every year, the time would come
when the number of the living at any age in that table
would express the number alive at that age in the
society in question. ‘Thus the number of persons aged
25 would be the 5879 survivors of those who were
born 25 years ago; and the number of the living
at every age and upwards would be found by multi-
plying the number alive at that age by the mean
duration of life in the table in p.166. If, then, the
law of mortality of such a society were required, it
would be found written in the burial registers of any
one year. For the numbers of births and deaths being
equal, there would be found for each year 10,000
burials ; which, if the law of mortality were permanent,
would be found distributed among the different ages
according to the table. Hence the number, out of
10,000 born, who attain a given age, would be found by
adding the number buried after that age.

But let us now suppose a population uniformly
increasing from year to year, say at the rate of 2 per
cent. per annum; such a population would double
itself in 35 years; and the younger lives would always
exist in a greater proportion to the older ones than

N 2

180 ESSAY ON PROBABILITIES.

would be indicated by a correct table of mortality.
The burials at the younger of two ages would therefore
occur in too large a proportion to those at the older.
Suppose, for instance, that 350 deaths take place at
the age of 40—41, and 1200 at the age of 5—6;
we are not therefore to conclude, that out of 10,000
individuals born, the deaths at 40 and 5 would be as
350 to 1200: for since the population doubles itself
in 35 years, those who now die aged 5, are part of
twice as great a number of such lives as were of the
same age 35 years ago: consequently, of the set from
whom 350 died at the age of 40, 600 died at the age
of 5. If, then, a table were constructed from burials
alone, without paying any attention to the rate of
increase of the population, the older lives would appear
too good of their kind ; that is, relatively to the younger
ones of the same society. This, as already observed, is
the case in the Northampton table; whereas, in the
formation of the Carlisle table, proper attention was
paid to the variation in question. The difference is
very perceptible in comparing each of these tables with
that of the insurance office which it most resembles.
At 25 years of age, the mean duration of the North-
ampton table is 30°9, and that of the Amicable 34:1.
If the proportions of the mean durations remained
nearly the same, (as generally happens,) then the
Amicable table at 60 giving 12°5, the Northampton
table should give 11°4; instead of which it gives 13:2.

The preceding supposes, that while the population
changes, the law of mortality remains stationary. It is
very unlikely that such should be the case; and observ-
ation, so far as it goes, tends to confirm the @ priori
suspicion. When provisions are cheap, or wages high,—
when, in fact,it is easy to maintain a family, — marriages
are more frequent, and are contracted at earlier ages. The
same abundanceof nourishment which tends to production,
also tends to preservation, both of parents and children ;
the consequence of which is, that a rapid increase of
population is often accompanied by a diminution of the

ON ANNUITIES. 18}

proportionate mortality. On the other hand, and from
contrary causes, a diminution of the rate of population
may be attended by an increase of the mortality.

As this work does not profess to enter further into
statistics than is necessary to exemplify the principles
of the theory of probabilities, I shall here close what I
have to say on the rate of mortality, considered inde-
pendently of the most important pecuniary applications.
The next chapter will point out in what way money
calculations are made.

CHAPTER IX.

ON ANNUITIES AND OTHER MONEY CONTINGENCIES.

Ir money could make no interest, the principles of
this chapter would be simplified, and the details
of calculation connected with it would be somewhat
reduced in amount. It will first be requisite to point
out the effect of compound interest, and to show how
to make computations connected with it. The funda-
mental calculation may be saved, for all such purposes
as this work is intended to answer, by the following
table, which may be described as follows. Opposite
to any year in the column headed Y, and under the
rate of interest in question (which is in numerals
at the head), will be found, within ten shillings, the
number of pounds which will, in such number of years,
at such rate of interest, produce a thousand pounds.
Thus, opposite to 23 years, in the column headed 3, we
see 507; that is, 5071. (or, more strictly, something
between 5106/. 10s. Od. and 517/. 10s. Od.,) will, when
improved at 3 per cent. for 23 years, produce 1000.
N 3

182

ESSAY ON PROBABILITIES.

y 2 2% 3 3% 4
IT 980 | 976" 1 O72 966 | 962
2| 961: | B52 4 948°) “934: 1/925
S| 942 |°929 | 915 | 902 | 889
4] 92 906 | 888 | 871 | 855
5 | 906 | 884 | 863 | 842 | 822
6 | 888 | 862 | 837 | 814 | 790
Z| S71 7841 | 818 | 661760
8} 853%) 821 | “789 | 78° | Yst
9! 887 | 801. | 766 | ‘794 |.708

10 | 820 | 781 | 744 | 709 | 676
11 | 804 | 762 | 722 | 685 | 650
12 | 788 | 744 | 701 | 662 | 625
18). 778 | 725 | 681 | 689 | 601
14| 758 | 708 | 661 | 618 | 577
15 | 748 | 690 | 642 j 597 | 555
16 | 728 | 674 | 623 | 577 | 534
M7} 74 + GST | 60S | 557. 151s
18 | 700 | 641 | 587 | 538 | 494
19 | 686 ; 626 | 570 | 520 | 475
20] 678 | 610 | 554 | 508 | 456

21 | 660 | 595 | 538 | 486 | 439

22:| 647 | 581 | 522 | 469 | 422

28 | 634 | 567 | 507 | 453 | 406

24 | 622 | 553 | 492 | 438 | 390

25 | 610 | 589 | 478 | 428 | $75

26 | 598 | 526 | 464 | 409 | 361

27 | 586 | 518 | 450 | 395 | 347

28 | 574 | 501 | 487 | 882 | 333

29 | 563 | 489 | 424 | 369 | 321

$0 | 552 | 477 | 412 | $56 | 308

$5 | 500 | 421 | 355 | 300 | 253

40 | 458 | 8372 | 307 | 253 | 208
a5 }.410°)' $29- | 264: )°:218 | 171
SO) S72°.| 291. | 228 | -179 | 141
Oe doer | 257 | 897.1. 151 | ile
.1.S05 | 297 | 3170 | 327 | 098
oi 276 | 201 | 3146 | 107° | Ore
70 | 250 | 178 ; 126 | O90 | 064
75; 226 | 157. | 109 | 076 | 053
80 | 205 | 139 | 094 | 064 | 043
85} 186 | 123 | O81 | 054 | 036
90; 168 | 108 | O70 | 045 | 029
95 | 152 | 096 | O60 | O38 | 024
JOO 1388 | O85 | 052 | 082° 1 020

ON ANNUITIES. 183

Ad 5 6 7 8 9 10 ye:
1 952 943 935 926 917 909 1
2 907 890 873 857 842 826 2
8 864 840 816 794 772 751 $
4 823 792 763 735 708 683 4
5 784 747 713 681 650 621 5
6 746 705 666 630 596 564 6
7 711 665 623 583 547 513 x
8 677 627 582 540 502 467 8
9 645 592 544 500 460 424 9
10 614 558 508 463 422 386 10
ll 585 527 475 429 388 350 11
12 557 497 444 397 356 319 12
13 530 469 415 368 326 290 13
14 505 449 388 340 299 263 14
15 481 417 362 315 275 232 15
16 458 394 339 292 252 218 16
17 436 371 317 270 231 198 17
18 416 350 296 250 2t2 180 18
19 396 331 277 232 194-| 164 19
20 377 312 258 215 178 149 20
21 359 294 242 199 164 135 21
23 342 278 226 184 150 123 22
23 326 262 211 170 138 Lig 23
24 310 247 197 158 126 102 94
25 295 233 184 146 116 092 25
26 281 220 172 135 106 084 26
Q7 268 207 161 125 098 O76 21
28 255 196 150 116 090 069 28
29 243 185 14] 107 082 063 29
30 231 174 Fol 099 075 057 30
35 181 130 094 068 049 036 35
40 142 097 067 046 032 022 40
45 11] 73 048 031 021 014 45
50 087 054 034 021 013 009 50
55 068 04] 024 O15 009 005 55
60 054 030 O17 010 006 003 60
65 042 023 012 007 004 002 65
70 033 017 009 005 002 OO1 70
75 026 013 006 003 002 001 75
80 |} 020 009 004 002 001 000 80
85 O16 007 003 OO1 001 000 85
90 012 005 002 OO1 000 000 90
95 010 004 O02 001 000 OOO 95
100 008 003 OO1 OOO OOO 000 100

184 ESSAY ON PROBABILITIES. |

When 000 appears in the table, the sum requisite is less
than ten shillings. Thus, less than ten shillings im-
proved for 85 years at 10 per cent., will produce a
thousand pounds.

There are six primary results which will be neces-
sary.

1. The present value of 11., meaning that fraction of
a pound which will, when improved at interest, produce
17. in a certain number of years. Since the present
value of 1000/. is tabulated, that of 14. is virtually so,
and may be found by placing the decimal point before
any result of the preceding table. Thus, °458/. is the
fraction of a pound which will, when improved at 5 per
cent. compound interest for sixteen years, produce 1/.; or,
in technical language, the present value of 1/. due sixteen
years hence, at 5 per cent. To find the present value
for any year not in the table, multiply the present
values for any years in the table the sum of which is
the number of years in question. Thus, to find the
present value of 1/. due fifty-seven years hence, at 3 per
cent., multiply together ‘337 and ‘961, the present
values for fifty-five and two years, the result of which
is *324, the fraction of 1/. required.

2. The present value of a perpetuity of 11, or the
sum which will, when improved at interest, pay 14 at
the end of every 365 days from the present time, for
ever. The number of pounds required is found by
dividing 100/. by the rate of interest ; being, in fact,
nothing but the sum which will at interest produce 17,
per annum. The results are in the following table : —

Rate P.V. of Perp. | Rate P.V. of Perp. | Rate P.V. of Perp.
2 £50 4 £25 ij £14
2 =6—£40 44 £222 8 £12}
3 £331 5 £20 9 £113
38 £284 | 6° Leroy 10 me

Thus 20/. improved at 5 per cent. yields 12 a year for
ever, provided the first payment be due at the interval
of a year from the time of putting the money out at

Ln a ee ee ee ee oer eet

a Re ee ee ee

ON ANNUITIES. 185

interest. This is also said to be 20 years’ purchase,
meaning the sum which would buy at once the annual

: payments of twenty years.

8. The present value of an annuity certain of 11.
(so called to distinguish it from a life annuity.) By an
annuity of 1/. is meant, a right to receive 1/. at the
expiration of every complete year after the creation of
the annuity, which is said to commence a year before
any payment is made; (or a term before payment is made,
whether the term be yearly, half-yearly, &c.) Thus
an annuity certain of five years, commencing January 1,
1838, is paid on the first days of 1839, 1840, 1841,
1842, and 1843.

When an annuity is to be designated which makes
one payment immediately, I shall call it an annuity
due; and a perpetuity of which one payment is to
be made immediately, a perpetuity due. ‘Thus on
January 1, 1839, the preceding annuity becomes an
annuity due of four years, comprising an immediate
payment of 1/7. and a commencing annuity of four
years. To find the present value of an annuity of 1/.,
use the following

Ruts. Multiply the perpetuity (present value of
the perpetuity) by the excess of one pound over the
present value of the last payment. Thus, for an
annuity of five years, at 4 per cent, multiply 25 by the
excess of 1 over °822, or by ‘178; the result 1s
4°4.5i., nearly

4. The amount of 1/. at compound interest, meaning
the sum to which 1/. with its interest will amount in
any number of years. It is found by dividing 1 by
the present value of 1/7. due that number of years hence.
Thus 1/. in 25 years, at 6 per cent., will amount to
(1/.+ 233), or 4°29/.

5. The amount of an annuity at compound interest,
meaning the sum of which the annuitant would be
possessed immediately after receiving the last annual
payment, if he had made compound interest upon every
preceding payment. This amount is found by multi-

186 ESSAY ON PROBABILITIES.

plying the number of years’ purchase in the perpetuity
by the interest (not the amount) of one pound im-
proved during the existence of the annuity. Thus, at
6 per cent., and in 50 years, 1/. will become (11.-+--054),
or 18°52, so that the interest is 17°57. This multiplied
by 162, or 19°, gives 2927. The more correct answer
is 290°34/., but the error is not ten shillings in a hun-
dred pounds.

6. The present value of an annuity for a term of years,
to begin after the expiration of another term; tech-
nically called a deferred annuity. ‘To find this, multiply
the present value of such an annuity created immediately
by the present value of 1/. due at the end of the term
of deferment. Thus an annuity of 10/. for five years, at
four per cent., to commence at the end of twenty years,
is thus found: were it to commence immediately, it
would (see last page) be worth 4°45 x 107. or 44°5/., and
the present value of 1/. to be received twenty years hence
is 456. Multiply 44°6 by -456, which gives 20°3/., the
present value required.

Extensive tables of the results of the preceding pro-
cesses are given in all works on interest or annuities ;
the present treatise is not meant to contain more than
enough to enable the student to exercise himself in first
principles, or the proficient to obtain approximate re-
sults, when no more extensive work is at hand. I now
pass to the consideration of annuities in general.

Though the term annuity be generally understood as
meaning a sum of money paid yearly or half-yearly to
an individual, yet it is important that the reader should
consider the word as implying any sum of money paid
at fixed intervals, for any term, definite or indefinite
provided only that the payment can never be suspended
and resumed again. Thus l/. to be paid every year
for ten years provided A live as long, with an ad-
ditional condition that payment is to cease if C in the
term should die, B being alive at the time of C’s death,
is an annuity within the meaning of the word: nothing
can make the payment cease at all, without making it
cease altogether. But 1/. to be paid as long as A, B,

ON ANNUITIES. 187

and C re alive, tu cease when one dies, and to re-
commence when a second dies, ceasing finally with the
death of the third, isnot one annuity, but two annuities.
And by whatever name periodical payments, may be
called, whether rent, salary, rent-charge, interest, &c.
&c., they are considered in this subject as annuities.

An annuity may be in possession or in reversion.
In the first case, a payment will become due at the end
of a term after the creation of the annuity, unless, in
the mean time, one of the conditions on which payment
depends cease to exist. In the second, the annuity is
not to begin to be paid until some circumstances happen,
or cease to happen, which are named in the agreement.
Sometimes the term reversionary annuity implies that
there is a contingency in the time of its beginning ;
and an annuity not to commence till after a fixed time
has elapsed, would be called a deferred annuity. There
is, however, a little variation in the use of these words:
reversion sometimes implies a remainder of something
existing, while a deferred interest is one which does not
commence till a future time. Thus if B be to take an
annuity on his own life as soon as A is dead, A en-
joying it during his own life, B’s reversion is a certain
part of an annuity on his own life, reckoned from the
present time. But if no annuity at all were paid until
A’s death, then B’s interest might be called a deferred
annuity. No difference arises from these distinctions in
formule or calculations; and they are useful in de-
scribing the circumstances of complicated problems.

A reversion, and also a deferred annuity, may be
certain or contingent: and the same of a reversionary
or deferred fixed sum. Thus A may have B's estate
after his death, provided he survive B; or else A and
his heirs or assigns may have the same: in the first
case the reversion is contingent, in the second certain.
A life insurance may be of one or the other kind: thus
A may covenant for his executors to receive 100/. at
his death, in any case, or else if B should be then sur-
viving. And if, in return for such insurance, A should
engage to pay a yearly premium, making the first pay-

188 ESSAY ON PROBABILITIES.

\
ment (as is usually done) at the time of contracting,
then the premiums altogether constitute an annuity due
upon A’s life.

All problems relating to this subject might be solved
from first principles, as will be shown ; but, in such case,
even the most simple of them would be attended with
laborious calculation. Tables are therefore constructed
of such results as will most facilitate the solutions of all
problems: and these tables give the values of annuities
on single lives, and on two joint lives. The annuity
presumed is always 1/. per annum, from which the
value of any other annuity is found by simple multi-
plication.

By the value of an annuity is meant the sum which
must be paid down in order to enable the grantor of
the annuity to pay it as it becomes due during the
term of its continuance. If money made no interest,
the average duration of such sets of circumstances as
are conditions for the payment of the annuity would
immediately point out its value. For instance, accord-
ing to the Carlisle tables, persons aged 40 live, one
with another, 27-6 years. Now, remembering that
half a year was added (p. 163.), as being the part of
their last year which, one with another, they pass
through, but which will not (in our meaning of the
term annuity) entitle them to any payment, we see that
27-1 is the average number of payments which such
annuitants, aged 40, will receive; that is, one hundred
with another, 100 annuitants will receive 2710 pay-
ments. Consequently, money making no interest, each
of them must pay 27°10/. for a life annuity of 11.,
or 27°10 years’ purchase. But if each had _ been
entitled to his fraction of the sum which would have
become due had he lived to the end of the year, then
27°60 years’ purchase would have been necessary.

Let us now suppose money to make interest. It
has before now seemed, to more than one writer, that
the value of a life annuity must be the same as that of
an annuity certain during the average duration of the
life. Many of my readers will not see the fallacy of

ail eet

ON ANNUITIES. 189

supposing that, money making 4 per cent., 16°7/., the
value of an annuity certain for 28 years, must be but very
little more than the value of an annuity on a life aged
40. Such a rule, if it be absolutely true, must be true
in any extreme case, however physically impossible.
Suppose, then, that of 101 persons aged 40, one lives for
ever, and the rest die between the ages of 41 and 42:
whence 100 of them will receive only one payment,
and the remaining one will receive a perpetuity of 1/.
The present value of an annuity of 17. for one year, at
4 per cent., is ‘Q6/., and 96/. increased by the value
of 12. for ever, is 1217, the present value of all pay-
ments. If, then, it be not known which is to have the
perpetuity, 4°17, is what each should pay, or 1°21.
very nearly. This result is undoubtedly correct: but
the average life of the 101 persons is infinitely great,
since there is no number of years which is not ex-
ceeded by them all put together. Each, then, would
have to pay for a perpetuity, if the preceding fallacy
were admitted ; or all together would pay 25 x 1012,
that is 2525/., more than 200 times the truth. It is
true that, applied to any actually existing law of life,
the incorrect notion cannot produce results so grossly
false as the preceding: but it is quite sufficient for
us to know that a rule is incorrect in its principle, to

| make it necessary to apply correct reasoning before we

can attempt to say how far it is wrong.

The rule for calculating the real value of an annuity
is made up of a collection of individual cases, not
more complicated than the following. A is to receive
one pound ten years hence, if a halfpenny which is
to be thrown up give a head; what is the present
worth of his chance? If he were to receive the money

' immediately, in case of winning, its value, on the prin-

ciples in chapter V., would be °5/.: but the present
value of 1/. deferred for ten years at 4 per cent. is

+ *709/.: whence that of °5/. similarly deferred is *355/.

This simple process contains the method of valuing
| the sum which must. be now paid down to secure the
| tenth payment of an annuity to a life belonging to a

190 ESSAY ON FROBABILITIES.

class of which just one half die in ten years. A
repetition of a similar process for every year in which
the annuity may become payable gives the present
values of the other payments. The sum of all the
present values of the different payments is the present
value of the whole annuity. Thus if we take a very
old life, and suppose that of 10 alive at the present
time there will be left at the end of successive years,
7, 5, 3, 0, there are three possible payments of the
annuity, and the chances of having to make them are
ry, qo, and -3,. But the first, if made at all, is not
made for a year, and the second and third are not made
for two and three years. If then a, }, and ¢ be the
present values of 1/. to be received at the end of one,
two, and three years, the values of the several payments
are +‘; of a, =°, of b, and 3, of e, the sum of which is
the value of an annuity of 1/. on any one life of the kind
in question.

By the status of an annuity, I mean the state or
condition of things during the continuance of which
the annuity is to be paid. This status may be simple
or complicated: in the latter case the method of find-
ing the chances of its continuance or termination will
also be complicated ; but this does not affect the simple
rule by which those chances, when found, are made, in
conjunction with the rules of compound interest, to give
the value of the annuity. Thus the status in question
may be of a compound character, allowing of several
distinct changes. For instance, A is to enjoy an
annuity to the end of his life, unless B should die
before C, in which case it is to cease. This annuity will
be enjoyed as long as either of the following status exist.

A, B, and C all living.
A and B living, and C dead.
A living, and B and C dead, C having died first.

Tables of the values of annuities are constructed in
the case of single lives and two joint lives; that is to
say, the mere inspection of the tables will enable us to

ae ee ea
es -

ON ANNUITIES. 191

answer the questions — what is the value of an annuity
to be paid so long as A shall live, and what is the
value of an annuity to be paid so long as A and B
shall both be alive, but to cease when either of them
dies. The authorities upon the subject have been
already mentioned ; namely, Mr. Milne * for the Car-
lisle tables, and Dr. Price + and Mr. Morgan { for the
Northampton tables. Besides these we have the work
of Mr. Baily§, now completely out of print; in con-
nexion with which must be mentioned the treatises ||
on interest and on leases by the same author, which are
also out of print and scarce. The work of Mr. Davies {J
contains, in connexion with the Northampton tables, a
very considerable amount of results, not elsewhere pub-
lished. The work of Mr. Baily on insurances has

lately been translated into French.** The results of

the Carlisle and Northampton tables have been lately
published on one large sheet, by Mr. M‘Kean.tft All
the preceding are valuable works, and very much sur-
pass, in useful application, all that has been done in
any other country, or in all countries put together.

I now give such a brief abstract of the results of the
Carlisle and Northampton tables as is consistent with
the plan of this work, in regard to annuities upon
single lives, and upon two joint lives.

* A Treatise on the Valuation of Annuities and Assurances, &c. &c., by
Joshua Milne. London. Longman and Co. 1815.

¢ Observations on Reversionary Payments, &c. &c., by Richard Price,
D,D. F.R.S.; seventh edition, edited by William Morgan, F.R.S. London.
Cadell and Davies, 1812.

t The Principles and Doctrines of Assurances, Annuities, &c., by Wil-
liam Morgan, F.R.S. London. Longman and Co. 1821.

§ The Doctrine of Annuities and Insurances, &c., by Francis Baily.
London. Richardson, 1816.

The Doctrine of Interest and Annuities, &c., by Francis Baily. Lon-

don. Richardson, 1808.

Tables for the purchasing and renewing of Leases, &c., by Francis Baily.
London. Richardson, 1807, (second edition).
{ Tables of Life Contingencies, &c., by Griffith Davies. London. Long-

| man and Co., and Richardson, 1825.

** Théorie des Annuités Viagétres, &c., traduit de l’Anglais par Alfred
de Courcy. Paris. Bachelier, 1836.
t+ Practical Life Tables, by Alex. M‘Kean. London. Richardson, Butter-

-— worth, &c., 1837. The annutties on two lives are given for every difference

ot age ; whereas, in the works of Messrs. Morgan and Milne, they are enly
given for multiples of five years, and the rest must be supplied by intere

. _ polation. This sheet 1s a beautiful specimen of typography.

192 ESSAY ON PROBABILITIES.

Values of an Annuity of £10, or (with the last figure
made a decimal) of £1, upon a single Life.

Northampton. Carlisle.

Age.}S p. ¢./4 p. ¢.'5 p. c.|6 p. c.f3 p. c.|4 p. ¢.|5 p. ¢./6 p. c./Age,

oO 7 123 | 103 89 76 | 178 | 143°] 121 | 104 @)
5 | 205 | 172 | 148 | 130 | 237 | 196 | 166 | 1438 5

{100 O O 0 O 17 17 16 16 |100

According to this first table it appears, for instance,
that at the Northampton rate of mortality, the value
of an annuity upon a life aged 45, at 3 per cent., is
13°7 years purchase, or 13/. 14s. Od. for an annuity of 12,

137/. for an annuity of 10/., and 1370/. for one of 1001. 4

perannum. Or rather, since the table is correct within

one twentieth of a year’s purchase, we should say that a
the value of such an annuity is between 13°65 and —

13°75 years’ purchase. The same remarks apply to
the table of annuities on joint lives.

ON ANNUITIES. 193

Values of an annuity of £10, or (with the last figure
made a decimal) of £1, upon two joint lives.

Northampton. Carlisle. |

194 ESSAY ON PROBABILITIES.

| Northampton. Carlisle.

D>
[tie]
g

5120; 148 | 130] 116 | 104 | 187 | 161 | 140
10 | 25 | 147 | 130 | 116 | 105 | 182 | 158 | 139
15 | 30| 137 | 122 | 110 | 100 § 171 | 149 | 132
20| 35 | 197 | 114 | 104 | 95 § 160 | 141 | 196
25) 40) 119 | 107 | 98] 90 | 148 | 132 | 119
30{ 45/169 | 100} 91] 84 § 137 | 123 | 111
35} 50/ 99] 91] 84] 78 | 123 | 112 | 102
40/55; 89 | 82] 77 | 714 107 | 98 | 90
45160| 78 | 73 | 68] 644 91 78
50|65| 66,| 62] 59] 564 77] 72/1 68
55 | 70 51 | 49 | 47 60 54
60 42 | 40] 389| 37 45] 43] 41
65/80] 31] 30| 29] 28 § 35] 34] 33
70/85} 21] 20] 20] 19 f 2 | | of

30 | 55 93 86 75 § 111 | 102

35 | 60 82 77 72 67 87 81
40 | 65 | 70 66 62 59 80 75 70
45 | 70 57 55 52 50 61 58
50 | 75 | 45 43 41 50 48 46

15 | 45 | 117 | 106 89 § 144 | 129 | 116
20 | 50 | 105 89 82 7 130 | 118 | 107
25 | 55 | 95 88 81} 76 § 113 | 103 95
30 84 78 | 73 95 88 82
35 | 65 | 72 64 60 81 76 71

ON ANNUITIES, 195

Northampton. Carlisle.

5140 | 124 | 112 | 101 | 92 | 154 | 136 | 122 | 110 | 5| 40
10| 45 | 120 | 109 | 99 | 91 | 146 | 131 | 118 | 107 | 10 | 45
15| 50 | 108 | 99} 91 | 8& F 131 | 119 | 108 | 99 | 15 | 50
20/55 | 96 | 80 | 82 | 764114] 105 | 96 | 89 | 901 55
25|60 | 85 | 79! 74 | 69 f 97 | 89] 83 | 77 | 25| 60
30|65 | 73 | 68 | 64 | GL f 82 | 77] 72) 68 | 30| 65
35 | 70 57 | 54 | 51 § 66 | 62 | 59! 56 | 35! 70
4)|75 | 47 | 45 | 43 | 41 § 51 | 49 | 47 | 44 | 40) 75
45|80 | 34 | 33 | 32) 319 41 | 39] 38! 36 | 45 | 80
50/85 | 2% | 23 | 23 | 22 F— 30 | 29 | 28 | 98 | 50 | 85
55/90] 17 | 16) 16 | 16 | 23 | 22 | 22) 21 | 55| 90
5|45 | 116 | 105 | 96 | 88 | 144 | 129 | 116 | 105 | 5 | 45
10} 50 | 110 | 101 | 93 | 85 § 133 | 120 | 110 | 100 | 10 | 50
15/55} 99 | 91 | 84 | 78 § 115 | 105 15 | 55
20; 60 | 86 | 80] 75 | 70} 98 | 90 | 84 | 7 60

45|90 | 17 17 16 16 24, 93 22 92 90
5 | 55 97 89 83 77 #115 | 105 96 89 | 51 55
10 | 60 83 78 73 # 100 92 85 79 | 10! 60
15165 | 76 71 63 85 79 74 70 |15 | 65
20!'70 | 61 58 55 53 68 64 61 57 | 20 | 70

75 48 46 44, 42 53 50 48 46 75

196 ESSAY ON PROBABILITIES.

' Northampton. Carlisle.
Ages. Ages
3 4 5 6 3 + 5
5 | 70 61 58 55 52 67 64 50 5 | 70
10 | 75 50 47 45 44 54 51 49 46 | 10 | 75
15 | 80 36 34 33 42 41 39 388 | 15

20190! 171 17| 17| 164 2% | % | 23 20 | 90
5 | 80 421 401 381 37 | 5180
10 | 85 32 | 31 | 30} 99 | 10185
15 | 90 04 | 24 | 93 | 92 | 15 | go
5 85 | -T 31 {30 | 99 | 98 | 5 | 85
10 | 90 25 | 24 | 93 | 921101] 90

The preceding table contains the values of annuities
upon two lives, for all ages which are multiples of 5,
Thus, for the ages 25 and 40, look to that part of the
table in which the ages differ by 15 years, and there,
opposite to 25 40, will be found, under 4 per cent.,
152 in the Carlisle table and 107 in the Northampton.
That is, money making 4 per cent., an annuity of 101,
which is to continue as long as lives of 25 and 40 are
both in being, is worth something between 1317. 10s. Od.
and 132/. 10s. Od. according to the Carlisle tables, and
something between 106/. 10s. Od. and 107/. 10s. Od.
according to the Northampton tables.

When the two required lives have ages which do
not end with the figures 0 or 5, proceed as follows: — Let
the value of annuity be required on joint lives of 38
and 47, (Carlisle tables at 3 per cent.). First take 35 and
45,and40and 45, and between the corresponding annuities
insert such a mean as would represent 38 and 45 upon the
supposition uniformly diminishing values. Then be-
tween 35 and 50 and 40 and 50 insert such a mean
as answers to 38 and 50. MHaving then 38 and 45
and 38 and 50, find such a mean as answers to 38 and
47. This process, which will be intelligible to a
reader who has practised similar ones before, will only

ON ANNUITIES. 197

be comprehended (if at all) by others from the
example.

35 and 45 133 35 and 50 123

40 and 45 129 40 and 50 120

4 3

3 3

5) 12 ey

2 2g

38 and 45 131 38 and 50 121
38 and 50 121
10

2 38 and 47 127 Ans

5) 20
4

Before proceeding further, I shall describe the
notation of which I intend to make use. It was not
the practice of the earlier writers to invent any dis-
tinctive notation of different contingencies, the first
attempt at which is found in the work of Mr. Baily.
Here, however, it was not carried to the full extent, and
Mr. Milne endeavoured to organise a system which
should take in every case, in which he succeeded perfectly
as far as distinct representation of all the cases which
occur. His symbols, however, are complicated and
strange, though I am clearly of opinion that they are .
much preferable to the attempt to dispense with
notation altogether. The new principle which the
notation I now propose involves, lies in the treatment
of terms of years certain as lives not subject to con-
tingencies. Thus, if AB represent an annuity on the
joint lives of A and B, meaning that it is to cease
when either A or B dies, then ¢B may represent an
annuity on the joint term of ¢ years and B’s life, to
cease with the first which expires ; or what would be

0 3

198 ESSAY ON PROBABILITIES.

called a temporary annuity on the life of B to last ¢
years, provided B should live ¢ years.

1. Any simple status on the existence or termination
of which a benefit depends, is denoted by juxtaposition
of large and small letters, the large letters denoting
separately the values of annuities on given lives or status,
the small'letters (or rather certain small letters, m, n, ¢,
for the most part) denoting given terms of years.
Thus ABC¢ is in existence as long as ¢ years last,
provided A, B, and C (or the persons on whom
annuities now granted have these values), remain alive
all the time.

2. A compound status, or one which exists as long
as either of two or more status remain, is denoted by
colons placed between the symbols of the simple status ;
thus A: B:¢ is in existence as long as A, or B, or ¢
years, any or all, are in existence. ‘The symbol is un-
meaning between two certainties: thus, n:n+¢ is
n+t.

3. A bar placed over two status indicates that the
one is to succeed the other and that the compound
symbol denotes a status in being as long as the one, or the

other after it shall exist: thus AP denotes a status
which remains in being during the life of A, and also
during a life to be named at the end of the year in
which A dies, having then the value P. If there be
occasion, a thicker bar, or one with an accent, or a
double bar, may be used where there are two successions
involved, between which it is necessary to distinguish.
4. The symbol | in all cases gives the whole symbol
the meaning of the present value of a benefit to be re-
ceived, and a figure attached, as in |,, denotes the rate
per cent. at which the value is to be calculated. This
benefit is always 1/7. ateach one or more payments. The
description of the status which must end before the benefit
begins will be found on the left of | ; and the description
of the status during which the annual payments of the
said benefit are to last will be found on the right. It
is further to be understood that the first payment made

ON ANNUITIES. 199

under A|B will take place at the end of the year in
which A drops, provided B be then in existence: thus

A|1 denotes * the present value Tt of an annuity of 1/,
the first (and only) payment of which is to be made at
the end of the year in which A drops ; while A|1 denotes
the present value of 1/., to be paid in a year from this
time, if A be then dead.

5. The last moment of a term certain is a part of
that term, unless the contrary be expressed by symbols:

thus 6|1 refers to a pound payable at the end of seven
years, and is 6|7. But when the last moment of a
term is considered as having followed the end of the
term, a small hyphen (considered as an abridged nega-
tive sign) is placed after the term in question: thus,

t-|n denotes an annuity of » payments, the first of
which is to be made at the end of ¢ years; and is the

same with t—I1|n.

6. The absence of symbols on the left of | indicates
that the first payment takes place in a year from the
present time; but -| indicates an annuity now due.
The absence of symbols on the right of | indicates
a perpetuity in reversion ; and | itself indicates a perpe-
tuity, the first payment of which is to be: made in a
year ; while -| indicates a perpetuity now due.

7. Dots between two symbols of status indicate
that the joint status shall be held to exist throughout
the year in which the first is determined, provided
the second remain at the end of the year; and
dots placed under several status denote that the
succeeding benefit is not due unless all those status
shall drop in the same year: thus |A...B denotes an
annuity on the joint lives of A and B, payable also
at the end of the year in which A dies, if B be then

alive; and A: Bil is the value of 17. at the end of the
life of the longest survivor of A and B, provided they

* One bar may be omitted in very simple cases.
t Remember particularly that in Ali, 1 means one year, not one pound.

Oo 4

200 ESSAY ON PROBABILITIES.

both die in the same year; also |AB... denotes the
value of an annuity on the joint lives to be paid in
addition at the end of the year in which the joint
existence fails.

8. When the condition is that a given status shall be
in existence at the moment in which another status
drops (whether the first last to the end of the year or

not), single dots are placed over the two: thus All B
means the present value of 1/., to be received at the end
of the year in which A dies, provided B be alive at that

moment ; while A|1B means the same, provided B be
alive when the payment is to be made.

9g. When it is a condition that deaths are to happen
in a specified order, it must be represented by writing

small figures under the status. Thus, A: B|C means ~
Lie

an annuity on the life of C, to begin at the end
of the year in which B dies, provided A have died

before B ; and A: (BC)|1 denotes the present value of
1

1/. payable at the end of the year in which the longest
of the two status A and BC drops, provided that the
status BC is determined by the death of B.

10. When the joint existence of one number of lives,
out of a larger number, is a condition, a figure may be

annexed as follows: (ABCD), indicates a status which

exists as long as any two out of the four are alive.

11. The double sign || indicates the premium which
is to be paid during the continuance of the status on
the left, in consideration of the deferred benefit described
on the right; premium being always interpreted as an
annuity due. And where a specific event, as distin-
guished from the duration of a status, is a condition, the
premium is to be held payable as long as any status exists
out of which that event may happen. Thus A: BC

is

denotes the premium which should be paid as long as
A lives with B, or B after A, to secure an annuity to
C when both are dead, provided A die first.

>

ON ANNUITIES. 201

* now proceed to some further instances :
| means the present value of an annuity of 1/. to last
m years.

m-|n the present value of an annuity of 1/., which is to
commence payment at the end of m years, and then
to last n—1 years, or n—1 more payments.

n| the present value of a perpetuity of 1/., com-
mencing at the end of n years, or first paid at the
end of m+ 1 years.

n-| the present value of a perpetuity, first paid at
the end of n years.

|A the present value of an annuity on the life of A.

|AB the present value of an annuity on the joint
lives of A and B, to cease with either.

t |A the present value of an annuity on the life of A,
to begin in ¢ years; that is, the first payment to
be made at the end of ¢+1 years, if A should then
be alive.

|A¢ the present value of an annuity on A’s life, or
t years, whichever drops first.

A| the present value of 1/. for ever, to be first received
‘at the end of the year in which A dies.

AB| the present value of 1/. for ever, to be first re-
ceived at the end of the first year in which A or B
dies.

A|t the value of an annuity for ¢ years, payment to
begin at the end of the year in which A dies.

AB|C the present value of an annuity which begins
payment at the end of the year in which either
A or B dies (the first), provided C be then alive,
and which continues during the life of C.

|A : B signifies the present value of an annuity which
is to be naid as long as either A or B is alive.

AB: C|D. Ethe present value of an annuity which is
not to be paid as long as A and B are both alive,
nor as long as C is alive, but which begins when
the joint existence of A and B, and that of C, are
both terminated ; and continues as long as either
D or E are alive.

202 ESSAY ON PROBABILITIES.

Brackets, as distinguished from colons, will serve the
same purpose as in algebra, namely, to give compound
terms the meaning of single ones. Thus,

(AB) C|D: E denotes the last-mentioned annuity,
on the supposition that payment is to begin when either
of two events happens, the failure of the joint existence
of A and B, or that of C.

AB|A: B the present value of an annuity to begin
when one of the two, A and B, dies, and to continue
during the life of the survivor, There is in this par-
ticular case the expression of an event which cannot
happen; for if B die first, it is only A who can receive
the annuity. Thus B|B is an expression for nothing ;
for the present value of an annuity on the life of B, to
begin at the death of B, is nothing. In the expressicn
AB|A: B, part is nothing, and the rest has a value.

A\l is the present value of 1/. to be received at the
end of the year in which A dies; ¢-|1 is the present

value of 1/. due ¢ years hence; ABJ|1C is the pre-
sent value of an annuity which, commencing with the
first death out of the two, A and B, lasts till either one
payment, or the life of C drops: that is to say, the
value of 1/. to be received at the end of the year in
which the joint existence of A and B fails, provided C
be then alive.

Let a colon placed after the final letter denote that
a perpetuity is one of the status during which the
annuity is to last. Thus,

A|(C:) denotes the present value of an annuity to
last for ever, after the death of A. The symbol denotes
that C being alive at the time of the first payment is a
necessary condition. ‘This being satisfied, the longest
of the two, C, or a perpetuity, is of course a perpetuity.

The presence of the colon always indicates the
longest of the two status, and when the colon is a final
symbol, one of the status is an infinite number of yet,
or 4 “igh it

A,:B,|C denotes the present value of an annuity

ON ANNUITIES. 203

which is to be paid during the life of C, after the
deaths of A and B, if A die before B.

es B: C, 1 Eis the present value of 10. to be paid

at the ad of the year in which the last of the lives,
A, B,C, drops, on condition that B shall have died second
or third, and that Hi shall be alive.

‘AIP denotes the present value of an arinuity which
is to begin payment at the end of the year in which A
dies, and to last during the life which shall then have
the value P. If there be several conditions, put a symbol
over the status which ends and before the one which
begins. Thus,

|ABC: P denotes the value of an annuity which is
to be paid during the joint existence of A, B, and C,
and further during that of a life which is to be
nominated (then having the value P) at the end of
the year in which either of the three drops: while

ABO(|P denotes that part of the annuity which is paid

during the life of P. An author’s interest in his
works, which is now denoted by |A : 28 was proposed,
in a bill lately before the House of Commons, to be

changed into A: 60.

The preceding notation will admit of almost any
degree of extension, and will be found perfectly capable
of expressing any case which now occurs in practice.
It must receive some generalisation before it can be
applied to an indefinite number of lives, in the manner
of Mr. Milne. But since it very rarely happens that
more than three lives occur in a practical question, I
shall leave farther extension to those who may find the
want of it. I shall merely now add, that any case

'- must admit of expression by means of a notation which

provides for the conditions under which the benefit
begins, the number of years which it is to last, and the
conditions of discontinuance: and also that in problems
of any degree of complexity, the invention of the
notation will be a useful preliminary to the actual solu-

204 ESSAY ON PROBABILITIES.

tion of the problem. Thus, suppose it required to
represent the value of an annuity which is to continue
as long as any two of the three, A, B, and C are alive.
This must be done thus :
[AB: AC: BC

This might be abbreviated into | (ABC),, or any such
symbol; which, however, I should recommend to no
one who is not very familiar with the developed form.

A method of making the notation of chances analogous
to that of annuities was devised by Mr. Milne, which is
much too ingenious and efficient to allow of its being
dispensed with in any future system. To adapt the
principle of this connexion to the system which I have
proposed, let nfa express the chance that A shall be
alive in m years, in which the small letters answering to
the capitals which denote lives refer to the chances
of those lives, and certain letters m, n,t (or more
if necessary), are reserved to signify terms of years.
Then all which precedes the sign t (or any other which
may be preferred), refers to lives or terms expired, and
those which follow the sign, to lives or terms which
are to be then in existence. Where no given term of
years is included, the chance must be understood to
refer to the whole continuance of all the status mentioned.
Thus,

atb is the chance that A shall die before B.

antb is the chance that one of the two, A or n years,
shall fail before B.

a:ntb is the chance that both A and n years shall
fail before B; that is, that B shall outlive A, and also
live more than n years.

atn the chance that A shall not survive n years.

a:ntb the chance that A shall die in n years, and

that B shall outlive that term.
ntab the chance that the joint existence of A and
B (or both A and B) shall outlast n years. ,
nm: a+ (n+1) the chance that A shall die in the

(n+ 1)th year from this time.

ON ANNUITIES. 205

The tables, of which an abstract has been given,
will enable us to find |A and |AB, the values of an
annuity on one or two lives, by simple inspection or
easy interpolation, but not |ABC, the value of an
annuity on three joint lives. A sufficient approximation
to |ABC is found by finding the single life Z, whose
annuity is equal, or as nearly equal as the tables will
give, to the annuity on the joint lives of the two elder
of A, B, and C. This new life must be placed instead
of those of B and C, which reduces the three lives to
two. That is |Z being equal to |BC (A being the
youngest of the three) |A BC is |A Z very nearly. Thus,
the ages of A, B, and C being 45, 50, and 65, we
have |BC (Carlisle table, 5 per cent.) =6°8, and in
the table of single lives 7°8 and 6:3 are the values of a
single life at 65 and 70 years, giving a diminution of
1:5 in five years or ‘3 per year of age. Hence 7°8
will become 6°8 in about three years, or 68 is the age
of Z. Again, writing the ages at the feet of the letters
signifying lives, |A,, Z,, is7°0 and |A,, Z,, is 5°8,
giving a diminution of 24 for every year of Z’s age, so
that |A,, Z,, will be 7°0—-7 or 6°3, which is near
the value of |A,, B,, C,;.

The values of annuities upon single or joint lives
being thus found, it is easy to solve many other pro-
blems of annuities. Such annuities may either be re-
quired for a temporary purpose, or as a provision for
future years, or for some parties after the death of others.
I write the result required in symbols at the beginning
of each question, and leave the reader to refresh his
memory of them by observing the demonstration.

Prosiem. An annuity of 1/, on the life of A, aged n,
is to be bought, to begin payment at the end of ¢ years,
if A should live so long: required its value.

The question is proposed in the most usual form,
but a little change will facilitate its solution. The
right of A is evidently an annuity to begin in t—1i
years, should he live so long. And the rule is expressed
thus,

206 ESSAY ON PROBABILITIES.

t—1|An=(t—l}fan) X (t—2|1)X Ang 4-4
At the end of ¢—1 years, if A be then alive (of
which find the chance), he enters upon an annuity of
1/., being then n + ¢—1 years of age. The value of an
annuity at that age, multiplied by the chance of attain-
ing that age, and by the present value of 1/., to be
received ¢—1 years hence, is, on the principles ex-
plained in p. 189., the present value of the annuity.
Example n=45, t=11 (Carlisle tables 5 per cent.)
10ta,,=4973= +862 9/1 =614 | A,,=10°3
862 x 614 x 10°3=5°45 £5°45 Answer.
Prositem. An annuity of 1/. is granted to A, aged n,
for ¢ years, provided he live so long: what is its value P

|A,, > A,—t|A,

The last equation is obvious; for the whole annuity

on A’s life is made up of a contingent annuity for
¢ years, and a contingent annuity to commence payment

in ¢+1 years. Consequently, from the whole value of - ;

an annuity for A’s life subtract that of an annuity to
commence payment in ¢+1 years, if he should be then
alive, and the remainder is the value of an annuity for
t years, if he should live so long.

Example n=45, t=10
By last problem 10]A,,=5°5; |A,,=12°6.
Therefore |(A 10)=|A,,—10|A,,=7:1 £7:1 Answer.

Prosiem. To find A,, |B,, the value of an annuity
on the life of B, aged n, the first payment of which is
to be made at the end of the year in which the life of
A, aged m, fails. This is called a survivorship annuity,
since it can never be paid unless B survive A. To
give this annuity is evidently to give a complete annuity
to B, on condition that he shall restore it as long as A
is alive; that is,

A|B=|B-|AB;

or, from the value of an annuity on B’s life subtract
that of an annuity on the joint lives. Thus (Carlisle

tables 4 per cent.), the value of an annuity on the life

iv

ON ANNUITIES. 207

of B, aged 30, to commence after the death of A aged
35, is 16°9—13°5, or 3°4 years’ purchase.

Propuem. To find AB|A:B, the value of an
annuity on the life of the survivor, whichever it may
be, after the death of the other.

AB|A: B=|A+|B—2(|AB)

This is the same thing as giving an annuity to both,
on condition that both restore it as long as both are
alive. From the sum of the annuities, therefore, on
the lives of A and of B, subtract twice the value of an
annuity on the joint lives.

Prostem. To find ABC|AB:BC:CA, or
ABC|(ABC),, the present value of an annuity to
begin payment at the end of the year in which one of
the three dies, and to continue as long as both of
the other two are alive. Give each pair an annuity on
their joint existence, and withdraw all three annuities
as long as all three are alive.

ABC|(ABC),=|AB +|BC+|CA—3(|ABC)

If the annuity be to commence immediately, without
waiting for the death of one, this is an additional grant
of |ABC or

|AB: BC: CA=|AB+|BC +|CA—2 (ABC)

Prostem. To determine |A:B, the present value
of an annuity to be continued as long as either A or B
shall be alive. Give both an annuity, but withdraw it
from one as long as both shall be alive.

|A: B=|A+|B-|AB
Propiem. To determine |A:B:C, the present

value of an annuity to be continued so long as any one
of the three shall be alive. Give each an annuity, and

_ withdraw one of the annuities from any pair as long as

that pair shall be both alive: but as this would take
away the annuity during the joint continuance of the

208 ESSAY ON PROBABILITIES.

three lives, grant an additional annuity of 1/. on that
joint continuance. Thus,

]A: B: C=|A+|B+|C—|AB-—|BC—|CA+|ABC;
in the process of finding which it becomes evident that
ABC|A: B: C=|A+|B+|C—|AB-|BC-|CA

It may be worth while to point out how, in every pos-
sible case, the preceding grants and withdrawals produce
the required effect ; namely, one annuity to be paid as
long as any one life lasts. The following are the pay-
ments and withdrawals : —

; le

Living. Dead. Pay. Withdraw. Balance,
ABC - - | 1+141+41 1+1+1 1
AB C 1+1 I 1
BC A 1+1 1 1
CA B 1+1 1 1
A BC 1 O 1
B AC 1 O 1
C BA 1 e) 1

Prosiem. Required C|AB, the value of an annuity
on the joint lives of A and B as long as they shall sur-
vive C. Grant a complete annuity on the joint lives, and
withdraw it while all three are alive.

C|AB=|AB-|ABC

Prosiem. Required A BIC, the value of an annuity
on the life of C, to commence after the failure of the
joint existence of A and B. Grant an annuity on the
life of C, and withdraw it as long as all three are alive.
Thus,

AB|C=|C—|ABC

Propiem. Required the value of an annuity B: C\A,
to be paid to A after B and © are both dead. Grant

ON ANNUITIES. 209

to A an annuity on his own life, withdrawing it as long

as either B or C survives with A ; that is, withdrawing

|AB: AC or |AB+|AC—|ABC. Hence,
B: C|A=|A—|AB—|AC+|ABC

Propiem. Required C|A: B, the value of an annuity
to be paid as long as either A or B shall survive C.
Grant one annuity to be paid as long as A or B is alive
(|A: B), and withdraw it as long as A or B lives with
(©; that is, withdraw |AC: BC. Hence,

C|A: B=|A: B-]AC: BC

=|A+|B-|AB-(|AC+|BC-—|ABC)
=]A+]B-(j/AB+]AC+|BC)+|ABC

The same case also amounts to granting |A: B: C and
withdrawing |C.

Prosiem. An annuity C|A: B, payable as long as
either A or B shall survive C, is to be divided equally
between them, while they both live, and is then to go to
the survivor. What is the value of the interest of
each ?

The interest of A is an annuity of half a pound
while both A and B survive C, and of a whole pound
as long as he shall survive both C and B: or
4(C|AB)+C: BiA. But,

3 (C|AB)=§ (|AB)—$ (|ABC)
: C: BJA=|A—|AB-—|AC+|ABC
the sum of which is

[A-§ ([AB)—|AC +3 (|ABC).

and similarly, the value of B’s interest is

|B-4(|AB)—|BC+§ (ABC)

_ the sum of which makes up, as it should do, the value
of C|A B_ »given above. :

P

210 ESSAY ON PROBABILITIES.

Proptem. An annuity on the longest survivor of |
A and B, or |A: B, is to be equally divided between them —
during their joint lives, and afterwards to go to the —
survivor. What is the value of the interest cf each? —
That of A is evidently 4 (|AB)+ B|A, which is j

4 ({|AB)+|A—|AB or |A—§ ({AB)
Similarly that of Bis |B-§(|AB)

which results are obtainable by a yet more evident
process, since the interest of each is evidently an annuity ©
on his own life, with half of it withdrawn as long as ©
both are alive. | a

Prostem. To determine (BC)|A, the value of an ©

annuity on the life of A, to commence with the failure —
of the joint existence of B and C, provided it be B who ©
dies first. a

There are no tables for the accurate solution of this ©
problem ; but the following reasoning leads to a result 4
which cannot be far wrong, unless some of the lives be~
very old, and which will always be near enough for the ©
species of application contemplated in this work. The ~
interest of A may be divided into two annuities, one ~
of which is B|AC, and the other a portion of B: C|A. 7
For A is certain of an annuity during C’s life after the ©
death of B, and of another after both are dead, provided ~
B die first. Suppose A’s interest in the latter annuity ~
to be worth one half of it, which is strictly true if B
and C be of the same age, and not much beside the
truth for considerable differences of age, particularly
when A is the oldest of the three. On this supposition
A’s interest is |

B|AC+§ (B: C/A)
or |AC—|ACB +4 (|A— |AB— [AC +|ABC)
or $ ([A—|AB +|AC—|ABC) =(BC)|A

which is obtained by supposing
} (A-|AB-|AC+!ABC)=B:CjA
1

ON ANNUITIES. 211

The facility with which the preceding rules may be
applied, in every part of the process except that of
finding the annuities on three lives, makes it unnecessary
to present examples. In order that examples may be
readily obtained in cases involving three lives, some
partial tables are given both by Mr. Morgan and Mr.
Milne, from which the following selection is made : —

Values of an Annuity of £10 (or, with the last figure
made a decimal, of £1,) on the joint continuance
of three lives of equal ages, from the Northampton
and Carlisle tables ; in the former at 4 per cent., and
in the latter at 5 per cent.

r=] S
5 23 2 g ae iT ee
Eg. a8 ce Eg ES ce
Eq So ao B< a8 a8
- ° an das ak Se |
| A Z
5 112 129 45 71 88
10 122 134 50 63 79
15 113 127 55 56 65
20 103 122 60 48 51
25 98 FIS 65 39 42
30 92 108 70 30 32
35 86 102 75 21 21
40 79 94 80 14 16
The following is from the Northampton table, at

4 per cent., the annuity being £10, and the ages as
specified.

Ages. Annuity. Ages, Annuity.

5, 15, 26 107 45,55, 65 51
10, 20, 30 104 50, 60, 70 42
£5, 25, 35 97 55, 65, 75 33
20, 30, 40 90 60, 70, 80 24
25; 35, 45 83 65, 75, 85 16
30, 40, 50 76 “70, 80, 90 11
35, 45, 55 68 75, 85, 95 2
40, 50, 60 60 °

p 2

212 ESSAY ON PROBABILITIES.

The following is from the Carlisle table, at 5 per
cent: —

Ages. Annuity. Ages. Annuity.

5, 30, 35 111 45, 70, “75 33
10, 35, 40 106 50,75, 86 25
15, 40, 45 99 55, 80, - 85 18
20, 45, 50 91 60, 85, 90 12 a
25, 50, 55 80 65,90, 95 11 |
30, 55, 60 66 70, 95, 100 9 :
35, 60, 65 55
40, 65, 70 44

I shall proceed in the next chapter to consider the
methods of finding the value of reversionary interests,
including life insurances.

CHAPTER X.

ON THE VALUE OF REVERSIONS AND INSURANCES.

and the last, lies more in names and in the circumstances
under which they occur for solution, than in difference —
of methods, principles, or (according to the scheme
which I have suggested), even of notation. Every —
interest, the symbol of which has any thing preceding —
the |, is properly a reversion, being something of which —
the benefit is not to begin until the happening of some
event, or the determination of some existing status. :

It may be proper here again to remark, that all the
rules in the preceding chapter, though the status men-_
tioned are technically called /ives, are equally true for —
any species of circumstances, temporary or permanent, ©
certain or contingent. Thus an annuity for ¢ years, to

ON THE VALUE OF REVERSIONS. 213

begin after n years, signified by n|n+¢, or by nit, is de-
termined by the same formula as an annuity on the life
of B, to begin after the death of A, signified by A|B.

n|n+t=|n+t—|(n n+t) A|B=|B-—|AB

but in the first I write |n instead of |(n n+) since
the joint continuance of » and n~+¢ years must be n
years.

The only difficulty of this notation is the necessity of
remembering the distinction by which it appears whether
n\n +t means that a payment is to be made at the end

of the mth year or of the (n+1)th. In the case of

.
|

A|B the first payment is to be made at the end of the
year in which A dies, which is always intelligible,
since it is considered as an infinitely small probability
that A should die at the moment which divides two
years. But since the term of mn years does expire at
such a moment, the analogy which connects the symbols
of terms certain and contingencies points out no rule.
Let n| stand for a perpetuity of 1/., the first payment to
be due at the end of m+1 years, then n—1| stands for a
similar perpetuity due at the end of n years. But since
the introduction of a new figure may turn the attention
off n, the datum of the problem, and since analogy
does not require us to strike off a whole year from the
term, let n-| signify a perpetuity deferred for a term
(no matter how little) short of n years, that is, payable
in n years. And by analogy -|A will signify an annuity
due on the life of A. The symbol n-|n+¢ must be
treated as if it were
n—1|n+t orn—1|t+1

The symbol of a perpetuity of 1/. to commence
from the present time (that is, payment at the end of
a year), is simply |. This is found as in p.184.,
and by subtracting |A, the value of an annuity on the
life of A, we obtain | — | A, the value of the reversion
of a perpetuity, of which payment is made at the end
of the year in which A dies. At the end of that year,
the holder of the reversion will have in possession and

P 3

214 ESSAY ON PROBABILITIES.

expectation, an equivalent to -| or 1+], or the value —
of a perpetuity due. Consequently, since the present —
value of | +| to be received at the deathof Ais |—|A, ©
that of 12. will be found by dividing the latter by the
former: or, |

The present value of 1/. to be received at the end of
the year in which A dies, is found by subtracting the —
present value of'an annuity on the life of A from that
of a perpetuity, and dividing the remainder hy the
present value of a perpetuity due, or one year’s pur-
chase more than the present value of a perpetuity.

Exampue. (Northampton tables, 3 per cent.) What
is the value of 1/., to be received at the end of the
year in which a life of 36 shall fail ?

Perpetuity of £1 at 3 per cent. £33°3
Value of annuity on life of 30 £16'9

34°3) 16°4(-478

In the preceding rule any status may be substituted
for a single life, and the value of the annuity which is
to be paid as long as the status lasts is connected with
the present value of 1/. to be received at the end of
she year in which the status fails, by the preceding
timple rule.

The premium which should be paid (first down, and
afterwards at the end of each year), is an annuity due
upon the life or status, and is therefore worth -| A or
1+ |A year’s purchase. Consequently the premium which
should be paid for the 1/. above described is the pre-
ceding present value divided by one year’s purchase more
than the annuity is worth. In the example, divide *4'78
by 1+16°9 or 17:9, which gives ‘0267, so that 2/7. 13s.
6d. is the premium for insuring 100/. at the end of the
year in which a life of 30 fails.

The following rule is somewhat shorter, in the case
in which the premium only is required, and not the
present value.

@uxstion. To find the premium which should be
paid (first down, &c.), during the continuance of a

ON THE VALUE OF REVERSIONS. 915

status, to insure 17. at the end of the year in which that
status drops.

Rute I. From the quotient of a perpetuity divided
by a perpetuity due, subtract that of an annuity on the
status divided by an annuity due.

34°3) 33°3 (‘9708 17°9) 16-9 (:9441
9441

0267 Answer, as before.

Rue II. Divide 1 by the value of an annuity due,
and by that of a perpetuity due, the difference of the
quotients is the premium required.

34°3)1(-0292 17°9)1(:0559
0292

‘0267 Answer, as before.

A perpetuity divided by a perpetuity due, is the
present value of 1/. to be received a year hence, and
may be taken from the following table : —

p.c p. c¢ p. c

2 ‘9804 44 ‘9569 9 ‘9174
ok ‘97.56 5 "9524 10 ‘9091
3 ‘9709 6 ‘9434

3h "9662 7 9436

4 ‘9615 8 ‘9259

From the preceding rule an illustration of the reason
of it may be derived, which I give professedly as an
exercise of ingenuity to those who may be beginners in
the subject. Let there be two persons, one of whom
holds a perpetuity and the other a life annuity, each of
1/7. Both the perpetuitant* and the annuitant desire

* If the holder of an annuity be an annuitant, the extension of lan-

guage is justifiable, by which the holder of a perpetuity may be calied a
perpetultant.
P 4

216 ESSAY ON PROBABILITIES.

to commute their interests for interests due: that is, the
perpetuitant, instead of 12. a year hence and so on,
desires to receive a fraction of a pound now, and the
same fraction at the end of every year; and the same
for the annuitant. Say the value of money is four per
cent., then the perpetuitant desires to change an interest
which is worth twenty-five years’ purchase into an

equivalent interest worth twenty-six years’ purchase a

(or income) ; consequently his year’s income (now due,
&c.) must be only $37. Say that the annuity is worth
ten years’ purchase; then by the same reasoning the
yearly income of the annuitant (now due, &c.) must
be only 447. The second is less than the first ; whereas
the original incomes were the same, both 1/2. But
there must be some consideration which the com-
mutation gives to the annuitant, and for which this
greater diminution of his income is the payment; and
it is as follows : —Since the commutation forestalls each
successive payment, giving it (or the substitute for it)
a year before it becomes due, the annuitant would
receive, if his income were made equal to that of the
perpetuitant, the 17. which, had he lived, would have
become due at the end of his last year, but which his death
hinders from becoming due. This difference of income

$2—10)/, is therefore equivalent to preventing his
receiving I7, at the end of the year in which he dies,
and it is taken from him now and in every succeeding
year of his life. Consequently it is the premium which
such an annuitant should pay to receive 1/. at the end
of the year in which he dies ; and it is also the result
of the first preceding rule.

The second rule may receive an explanation of a
similar kind. I now reverse the problem, and ask the
following |

Question. If an office charge the premium p for
insuring 1/. at the end of the year in which a life (or
other terminable status) drops, what should we infer
that they suppose to be the greatest possible value of
an annuity to continue during the remainder of that life

ON THE VALUE OF REVERSIONS. 217

or status; that is, what is the value of an annuity on
that status, which is such that the office must be ruined
if the truth falls below it?

Ruue. Take the rate of interest which money
really makes *, and subtract the premium for 1/. from
the present value of 1/. to be received at the end of a
year (see last table): divide the remainder by the
excess of unity over the remainder, and the quotient is
the number of years’ purchase in the present value of
the annuity. :

Exampie. A society professes to insure lives of
35 at a premium of 3 per cent. on the nominal sum
insured ; what is the lowest value of the annuity on
such lives at which this can be done ?

At 4 per cent. {1 = °'9615 1-0000
(page 200.), Al|1 = *0300 9315
0685) -9315(13°6 0685

Answer. Such an office cannot permanently stand
(as far as this one species of bargain is concerned),
unless the value of an annuity on lives of 35 (at 4 per
cent.) be more than 13:6 years’ purchase.

Generally speaking, contracts of insurance are not made
for the end of the year in which the party dies, but for
payment at a given number of months after the parties’
death is proved and the claim made. If this agreement
were always made for six months after the real death,
the office would, one party with another, neither gain
nor lose, while for every month less than six, the office
gives that month’s interest to the parties’ executor, while
for every month more than six by which payment is
deferred, the office takes a month’s interest. _I believe
no office defers its payments more than six months after
the claim is made ; and the difference is rendered im-
material by the probable errors of the tables, which
require too large a covering profit to make it worth
while to take such a circumstance into account.

* This is an essential element, but cannot be very accurately determined ¢
something above the truth should be assumed,

218 ESSAY ON PROBABILITIES.

The premium demanded by an office is that charged
by their tables at the age which the party will attain at
his next birthday ; thus if a person desire to insure his
life the day after he attains 31 years complete, he will
be required to pay the same as if he had deferred
completing the insurance till the day before his thirty-
second birthday... This is, one party with another, a
gain of half a year to the office. Thus, the North-
ampton table at 3 per cent. giving 16°7 and 16°5 as
the values of annuities at the above-mentioned ages, all
parties who have passed 31 years at their last birth day
are considered as having lives worth 16°5, whereas they
are worth, one with another, 16°6. The tables are not ~
sufficiently accurate to make the effect worth caring for.

A party having made an insurance, and paid one or
more premiums, the instrument by which the right to
receive the stipulated sum at death on payment of a
stipulated premium is conveyed, is called a policy of
insurance. The value of this policy is then easily
determined ; at least what we may call its office value,
‘supposing the tables of the office to be perfectly correct.
A person aged thirty insures for 100/., for which he pays,
say 3l.; he continues to pay this premium until the
age of fifty, at which time, if he had began to insure,
the annual premium would have been, say 5/. Suppose
that the holder of the policy wishes to sell his interest
just before he would otherwise have had to pay another
premium, it is plain that he then offers for an in.
surance on the life of 50, a better bargain than the
office would offer, since the buyer of the policy (who
pays all future premiums) will acquire, in consider.
ation of an annuity due of 37. upon the life of A, that
‘which the office would not sell for less than an annuity
due of 5/. upon the same life. The difference, or an
annuity due of 2/. upon the same life, is the value of
the policy.

Ruue. To find the present value of a policy of insur
ance, at the moment before a premium becomes due, sub-
tract the premium which is to be paid from the premium

|

ON THE VALUE OF REVERSIONS, 219

which would be paid if the same party made the same
insurance at the present time. Find the present value
of an annuity on the life of the party insured, of the
same yearly amount as the preceding difference, and this
value, increased by one year’s purchase, is the present
value of the policy.

To find the value of the policy immediately after
a premium is paid, add the premium just paid to the
result of the preceding rule. It would not be worth
while, in the present work, to give a rule for any in-
termediate value. (See Milne, p. 283.)

Bat in finding the real value of a policy, there are
one or two circumstances to be considered, of which no
mention is made in the preceding rule. The buyer of
the policy, being uncommitted by any previous act of
his own, is not bound to consider the premium of any
one office asastandard. Suppose that in the preceding
example, another office of equal solvency can be found,
which will insure a life of 50 at 4 per cent. instead

of 5: the buyer, therefore, may consider that the

seller offers him for 3/. a year during his life a benefit
which he might buy elsewhere for 4/., and that he
should therefore pay only the value of an annuity due
of 1/. instead of 27. But since the two offices cannot
be together parties to any transfer of policy, the pre-
ceding case will only serve to show that it may be
more prudent for a person who has money to invest, to
lay it out at once in insuring lives in a cheap office,
than in buying existing policies in a dear one. It is to
be remembered that the lower premium in the preceding
rule is to be paid, by bargain already existing, while
the higher one is hypothetical, depending on the

buyer's opinion of tables of mortality. That the office

which demanded and obtained the 37. would demand
the 5/. for an insurance now to commence, must be no
consideration for a person who is merely thinking how

to lay out his money to the best advantage ; it may be

by buying the policy which is offered to him, or by
insuring his own life, or that of some one else, in the

220 ESSAY ON PROBABILITIES.

same or another office. It is his business to consider
what he is likely to have to pay, in the shape of future
premiums, and not what an office, which must be on
the safe side, has thought fit to suppose it will have to
receive. Putting out of view the state of health * of
the party insured, I should think it most advisable to
calculate the value of policies by finding the present
value of the sum insured, and also that of the premiums
to be paid, from the tables which best represent healthy
life, and using the rate of interest which money will
really obtain, rather above than below ; that is, I should
use the Carlisle tables at 4 per cent. The profits
guaranteed by the office, if any, should be duly con-
sidered. Thus suppose a person at the age of 30 had
insured for 10007. in an office which demands 25/.
premium for that insurance, and returns no profits,
and suppose that twenty years have elapsed, so that the
life insured is now at the age of 50, what is the real
value of his policy? The value of 1/. to be received at
the death of a person aged fifty, by the Carlisle tables
at 4 per cent., is (p. 214.), 25—12°9 divided by 25+1
or *405: that of 10001. is therefore 4657. If a premium
be just becoming due, the present value of all the
premiums is therefore 1 -+.12°9 or 13:9 years’ purchase ;
and 13-9 x 251. is 347-51. Consequently 405 — 3474, or
1174/., is what J should consider to be the value of that
policy. But if I took the tables of the same office,
which require a premium of 47/. at the age of fifty,
and which, with some variation, are derived from the
Northampton tables at 3 per cent., I should find by
the rule in p.218., (14+12°4) x (47—25), or 2951.
nearly. So great is the difference between policies
valued by the nearest approximation which exists to the
actual truth, and then valued by the tables which
offices adopt for their own security.

The office itself, which takes an advantage of the
buyer when the policy is first created, may reasonably

* Of course the policy of a person whose health has very much declined
since he effected the insurance, is of higher value on that account; but
this cannot be made the subject of calculation.

ON THE VALUE OF REVERSIONS. 921

allow that advantage to the insured, if he afterwards
desire to sell his policy to the office itself. I am not
aware of the exact rule which is followed by the offices
in this respect, except in one or two cases, in which
the plan is, or was, to follow their own tables, with a
certain deduction from the result, and to give the dif-
ference to an insured party who desired to sell his
policy. This is well enough in the case of offices which
return profits; but if such a rule be followed by those
which do not, it may amount to a contradiction of their
profession in the case of the sale of policies; and may
become in effect an allowance of that share in the profits
to those who desire to leave the office, which they re-
fuse to grant to those who continue. To prevent such
a result, I believe the offices who would be liable to it,
make a large deduction from the value of policies, as in-
dicated by their tables.

All the preceding rules apply to any given status as
well as to a given life. Thus, to effect an insurance on
the survivor of two lives, the present value and the
premium (payable as long as either is alive) are to be
found by using |A+|B—|AB for the value of the an-
nuity, instead of |A. I now proceed to some simple
cases of insurance, where the payment on one party’s
death is made conditional upon another party being alive
to receive it.

The symbol A| (1B) denotes the value of an annuity
upon the joint continuance of one year and the life of B,
payment being made at the end of the year in which A
dies. It is therefore necessary that B should be alive
at the end of the year in which A dies. But in the
usual conditions of contingent insurances, it is sufficient
that B should be alive at the moment in which A dies.
Let this be expressed by A|1B; it is then evident that
A|1B is greater than A|1B. The following preliminary
considerations will be necessary.

Propiem.—Required the value of an annuity on the
joint lives of A and B, to be paid at every end of a
year at which B shall be alive, provided A were alive

229 ESSAY ON PROBABILITIES.

at the beginning of the year. This may be denoted
by | (A...B).

‘The condition that a life shall be alive at the begin-
ning of the year must be, in tables of averages, the
same as that of a life a year younger being alive at the
end of the year. For example, suppose that of 500
persons of the age of n—1, 493 attain to that of n,
then 500 annuities granted on the lives of Ay-, will
be equivalent to 493 granted to A,, if the latter be
payable at the end of any year in which A shall have
been alive at the beginning ; and the same for any
joint lives combined with both. Thus 500 annuities
granted on the joint lives of A,1 and B, payable as
usual, are equivalent to 493 on the joint lives of A,
and. B, payable as long as B is alive at the end, and A
was alive at the beginning, of ayear. Hence | (A...B) is
29° of |A|B, where A, stands for a life a year younger
than A. MHence the following Rune. To solve the
preceding question, multiply the value of a joint an-
nuity on Band one year younger than A by the number
alive at that younger age in the table, and divide by
the number alive at A’s age: the result is the present
value required. Or more concisely, divide | (A,B) by
the chance which A, has of living a year.

Now let us ask, by how much does |(A...B) as above
described exceed |AB. The only possible case in which
a payment will ever be made upon the first annuity, and
not upon the second, is when A dies before B, for both
are determined by the death of B. When A dies be-
fore B, the annuity will be paid at the end of the year
upon |(A....B), if B be alive, but not upon |AB. Con-
sequently the excess of the first over the second is the
value of £1 to be received at the end of the year in
which A dies, provided B be then alive.

Or,

AJIB=|(A...B)-IAB, Bl1A=|(B...A)—|AB

Also the present value of an annuity to be paid at the
end of any year in which both A and B were alive at the

ON THE VALUE OF REVERSIONS. 223

beginning, or |AB..., is evidently | AB increased by ABI1,
the present value of an insurance of one pound on the
joint lives.

Prosiem. Required the present value of £1 to be

|AB, on the joint lives of A and B.

|(AB...) the same as |AB, but to be also paid at
the end of the year in which the joint existence fails.

Take in exchange the following annuities : —

\(A...B) and |(B...A) two annuities to be paid
during the joint lives, and also at the end of the year
in which the joint existence fails, provided B in the first,
and A in the second, be alive at the end of the year.

The balance of this transaction will be £1 to be paid
at the end of any year, provided B and A both die in
the year. For as long as both are alive, two annuities
are payable each way; if A die and B remain alive
till the end of the year, |AB has ceased, |(AB...) is
payable, but |(B...A) has ceased, and (A...B) is payable ;
similarly in the case of B dying and A remaining alive.
If both die in one year |AB has ceased, but |(A B...)
is payable, while |B...A and |A...B haye both ceased.
Consequently, the only possible payment which the
grantor has to make, over and above those which he
receives, is the £1 in the question proposed ; or

We are now in a condition to solve the final Prosiem.
Required the value of #1 to be paid at the end of the
year in which A dies, if B should have been alive at the |

moment of A’s death. This is denoted by A\1B.
When A dies before B, either B survives till the
end of the year, or dies in the intermediate time. The

insurance on the first risk is worth AJ1B determined
in p. 222.; on the second it is worth half the result of
the last problem, if it be considered that the chances of

224 ESSAY UN PROBABILITIES.
A dying before and after B, in any one given year, are
equal. We have therefore

|A...B—|AB+4{|AB+|AB...—|A...B—|B...A}

Or,

I{|AB...-|AB+|A...B—|B...Af
But, (p. 223.) os
|AB...—|AB is AB]I
Whence the final result is,

4fABl1+|A...B-|B...A}

Rute. For determining the value of 1/. payable at ]
the end of the year of the death of A, provided B be
alive at the moment of A’s death.

(4.) From the value of a perpetuity subtract the q

value of the joint annuity, and divide by that of a per-
petuity due.

(2.) Multiply the value of the joint lives of B and a
year younger than A by the number alive in the table
at the younger age, and divide by the number alive at
the age of A.

(3.) Repeat the preceding process, substituting B
for A, and A for B.

(4.) From the sum of the results of (1) and (2)
subtract that of (3), and half the difference is the pre-
sent value required. The premium payable during the
joint lives is found by dividing the result by the value
of an annuity due on the joint lives.

Though I have deduced the preceding rule at length
as an instance of the very improving exercise of deducing
the more complicated results of this subject by what
we may call the balance of annuities, yet for the rough
purposes of this work, if not for others still more exact,
the more simple process implied in finding |A...B—|AB
is sufficient. It must be obvious that the fraction of
the whole value of a survivorship insurance, which
depends on the risk of both parties dying in the same
year, is a small one ; so that it is nearly sufficient (and
certainly within the probable errors of tables, &c.) to

ON THE VALUE OF REVERSIONS. 225

consider this question independently of the double risk,
- and as if it were certain that both parties would not die

in the same year. Instead of A]1B, we then employ
A\IB (page 222). Mr. Milne (in his page 352. &c.)
_ has given three examples, which I here repeat by the
_ latter formula, taking the data from the work cited:

1. Ais aged 17, B is aged 57: Carlisle Tab., 4 per cent.

[A... oe Ali B= ‘086, which in work cited
[AB= 9-923 [is 08656.
0:086

2. A is aged 45, B is aged 35; do. do. 5 per cent.

'A,.. B=11°172 — : ( : ‘
| AB=10°912 li B = °260, which in beg cited
0:260 [is *266331.

3. A is aged 64, B is aged 19; do. do. 5 per cent.

a om apres Alt B= °522, which in work cited
So [is-52556.

0°522

_ The present value of a pound, to be received at the
_end of the year in which A dies, provided he survives

B, or ‘Beal, is readily found from the preceding:
» ieee
for since A must either die before or after B, the sum

of the two must be the present value of 17. to be received
at the death of A, independently of B. That is,

raat! [=A |1—/

AliB = All _ All B very nearly,

The present value of an insurance is also that of the
reversion of a fixed sum ; since it is the same thing whe-
ther 1/. is to be received from an office, or conditionally

under a will, or in any other way. The reversion of a
s

fi
4

7

226 ESSAY ON PROBABILITIES.

perpetuity should be treated as that of the value of a :
perpetuity due at the end of the year in which the life ‘

drops.

Propiem. Required the present value of 12. to be re-
ceived at the end of the year in which A dies, provided —
that event take place before the expiration of ¢ years —

from the present time: signified by A| lt.

Suppose a person to have the certain reversion of a ©
perpetuity due at the end of ¢ years, or sooner, if A die —
before ¢ years are expired : the reversion of the perpetuity —
after the failure of the joint existence of A and ¢ years, ~
is |—| A t, which can be found from page 206. But —
this is more than that fraction of a perpetuity due at the —
end of the year in which A dies, which will pay for the |
chance of entering on it before ¢ years are expired: for ©
part of it expresses the value of a perpetuity due, which, —
though A should be alive, is to be entered on by the ©
failure of ¢ years. If ¢ta@ be the chance that A is ©
alive at the end of ¢ years, then tt a x ¢| must be ©
deducted, as being expressed twice in the preceding: —

consequently

|-|At—ttaxt|

is the present value of a perpetuity due, or -|, to be ©
entered upon at the end of the year in which A dies, if —
before ¢ years. The preceding then divided by -| gives —
the present value of 17. to be received under the condi- ~
tions of the question. But a perpetuity created at the ©
end of ¢ years, or ¢|, divided by a perpetuity now due, ©
gives the present value of 1/. to be received at the end of ©

t+1 years; which gives the following

Ruue. From the value of a perpetuity subtract that of —
an annuity on the given life for ¢ years, and divide by

the value of a perpetuity due. From the quotient sub-

tract. the present value of one pound to be received at —
the end of ¢+1 years, if the life be in being at the end —
of ¢ years: the difference is the present value of 1/, to —
be received at the end of the year in which the life drops, _
if before ¢ years have expired. The present value just —

we
SEN EAS a Se Sema

ane
ie

ef

as:

ON THE VALUE OF REVERSIONS. 227

found, divided by that of an annuity due on the given
life for ¢ years, gives the requisite premium.

And since the present value of such an insurance as
the preceding, together with the present value of 1/ to
be received at the end of the year in which A dies, if
after ¢ years, make up the present value of 1/. to be re-
ceived in any case at the death of A, the third diminished
by the first will give the second. But it will be prefer-
able to make an independent investigation of this case.

Prosiem. Required ¢: A 1 the present value of 1/.
1 2

to be received at the end of the year in which A dies,
provided ¢ years shall have previously expired.

If from the present value of a perpetuity deferred for
¢ years, we deduct that of an annuity on the life of A de-
‘erred for ¢ years, we have the value of a deferred perpe-
tuity, further suspended during the term by which A out-
lasts ¢ years, and to commence at the end of the year in
which A dies: or not to be suspended at allif A should
die in less than ¢ years, Take away the value of a per-
petuity beginning from the end of ¢ years, if A should
_ have died in the interval, and we have remaining the
_ present value of a perpetuity due at the end of the year
in which A dies, if that be deferred beyond ¢ years.
This last is therefore

t]}— ¢<|A —(1-tta) x7
where ¢ f a is the chance of A living ¢ years. This can
be reduced to
ttaxit|—tlA
Divide this by the value of a perpetuity due, and we
_ have the present value of 1/. receivable on the same con-

ditions, But ¢| divided by -] gives¢ | 1, as before;
whence the following

Ruz. Multiply the present value of 1/. receivable at
_ the end of ¢+1 years by the chance which A has of
_ living ¢ years; and from the product subtract the quotient
@ 2

228 ESSAY ON PROBABILITIES.

of a deferred annuity on A’s life, divided by a perpetuity
due: the remainder is the present value of 1/. to be re-
ceived, if A outlive ¢ years, at the end of the year in
which he dies.

The preceding pages contain all those cases which
most usually occur in practice; but it is to be noticed
that various modifying circumstances will present them-
selves in different problems, for which no general rule
can be given. I now proceed to problems in which
successions occur ; that is, in which a benefit depends
upon the continuance of a status which does not begin
until another status is finished.

In the case of an annuity for a certain term of years
t, to begin payment at the end of the year in which A
dies, we have obviously to consider a benefit which, at
the end of the said year, will amount to an annuity due,
of ¢ payments; or 1/. augmented by the present (as it
will be then) value of ¢—1 future payments. This
then value can be found as in page 185., and, being mul-
tiplied by the present value of 1/. receivable at the end of
the year in which A dies (found in page 214.), gives the
present value of this contingently deferred annuity.

In the case of A|B, an annuity on the life of B after
the death of A, we certainly have a succession, but it is
one which may never exist. To make a problem which
may come under the present division of our subject, we
must imagine that, at the end of the year in which
A dies, a new life may be nominated at pleasure, which ©
is then to be of a givenage. If P be the value of an ©
annuity upon such a life, then, according as the benefit |
is an annuity, or an annuity due at the end of A’s year ©
of death, we find the present value of P or 1+ P to be
received at the end of that year. The result is the pre- —
sent value of the succession. This problem includes —
that of finding the value of the next presentation to a
living. The patron of a living of 500/. a year may con= —
sider that he gives the clergyman whom he presents —
100/. a year (or whatever may be called liberal remuner-
ation for a curate) for work and labour, and the remain-

ON THE VALUE OF REVERSIONS. 299

ing 400/. as a free gift. If he sell the next presentation
he must therefore consider that he sells 400J. a year (not
500/., since that would be to allow the clergyman no
salary* for his labour), to be paid yearly during the con-
tinuance of a life to be named by the buyer, at the de-
cease of the present incumbent. And, since the right to
name new incumbents of 24 years of age is part of the
bargain, the patron will require a sum corresponding to
the value of an annuity upon a life of that age. De-
ducting a sum for first fruits, probability of expenses
from dilapidations, &c., which must be determined by
the circumstances of each case, the remainder is the net
present value of the living. It would probably be most
fair to value the interest of the purchaser as if the new
incumbent would come into half a year’s revenue at the
end of the year in which the present incumbent dies.

Question. What is the present value of the next
presentation to a living of which the average annual
income is £s, the salary of a curate t being £v, and f the
estimated expenses at entry. Let A be the value of the
incumbent's life, and P that of a life of 24 years of age.
Find the present value of P + 3, to be received at the end
of the year in which A dies (p. 214.), and multiply the
result by the excess of s over v; from this deduct the
present value of f, to be received at the end of the year
in which A dies, and the remainder is the net present
value of the next presentation.

The perpetual value of an advowson (that is, of the
right to nominate the incumbent in all time to come,
after the decease of the present one) is generally valued
as the reversion of the net income after the death of the
present incumbent. But the expenses of entry, first
fruits, &c., should be considered as a fine levied on the
property at the death of every tenant, in diminution of

* The right of selling livings is therefore a bond fide right to alienate
all the church property which is in private hands, with the exception only
of that minimum which will obtain a curate.

+ If the living be one on which a curate must be kept by the incumbent, |
the salary of two curates should be deducted from the yearly revenue in the
valuation.

e 3

230 ESSAY ON PROBABILITIES»

the total value. To the problems connected with this
subject I now pass.

A great many interests are held in this country on —
the consideration of rents, fines, or whatever they may —
be called, which are not paid at any fixed time, but at —
the deaths of successive lives which are named, each life ©
being nominated, and the rent or fine paid, at the death —
of ‘the preceding nominee. Leases held under ecclesi- ©
astical and other corporations, copyholds, &c., are in-
stances. By a statute of Henry VIII., corporations are ©
permitted to lease lands for three lives, or twenty-one —
years; so that it may be suspected the legislature ima- —
gined the average term cf the duration of three lives to ©
be 21 years; or that, any three mature lives being ©
named in one set, and a large number of such sets being ©
taken, and each set being considered as a status to last ©

as long as any one of its lives was in being, the average

duration of such a status was 21 years. If this were ©

the opinion, and grounded upon any thing like experi-

ence, the value of life in that day must have been incre- _
dibly below what it is at present: but it must be
remembered that in that day of insecurity few people ©
would venture on the life of a child or a woman; and —
that in all prebability the lives contemplated were those —
of men of middle age. However this may be, since that —
time the tenure of lease upon lives has become ex- —
tremely common, it being understood that the lives ©
which drop are renewable upon the payment of a fine, —

either fixed or at the discretion of the lessor.

It is, of course, the interest of the iessor that the lives :
should be as bad, and of the lessee that they should be ©

as good, as it is possible: but the lessee, having the no-
mination of the lives, will choose the best the tables
afford. The rate of interest being settled, the highest

life annuity in the tables gives the age which the life :
nominated ought to have. The best age in the North-

ampton tables is 8 years, and in the Carlisle 7 years:

for which ages J subjoin the values of annuities at va-

rious rates of interest, adding also the age of 24, which

ON THE VALUE OF REVERSIONS. 231

will be useful in the calculation of the values of advow-
sons, with the correction above proposed

Northampton. | Carlisle. |
——- j “
Ages.|3p.c.|4p.c.J/5p.c. BR edt 4p.c. |5p. c.|6p.c.|Ages.
8 | 209 17°7 | 15°2 | 133 23°9)} 198 | 168 1}145] 7
24 | 180 | 15°6 | 13°7 | 12:1, 20°9 | 17°83 | 154 | 135 | 24
|

Question. At the end of the year in which A dies,
a fine of 1/. is to be paid, and a new life nominated, of
which the value will then be P: at the end of the year
in which P dies, another fine of 12. is to be paid, and a
new life P nominated, and so on for ever: what is the
present value of all the fines, or what present money
must a person be considered as paying who receives an
estate charged with the preceding liabilities P

This problem, as will be more fully explained in the
second appendix, was incorrectly solved by every writer
on the subject, down to the time of Mr. Milne, whose
solution, though perfectly correct, is in a difficult form.
The coincidence of the rule I now give with that of
Mr. Milne will be shown in the appendix cited.

Let us suppose a fine of 1/. per annum, first payable
at the end of the year in which A dies. If, then, a
receiver P were appointed for his life, his interest in the
fines, at the end of the year in which A dies, would be
1+ |P; and if at his death a second receiver were ap-
pointed, of the same age at which the first was when his
term began, the interest of this second receiver at his
entrance would also be 1+|P, and so on. But if the
tenant compounded with each receiver on his entrance,
for the rents payable during the life of that receiver, it
would evidently be equivalent to paying a fine of 1+|P
at the end of the year in which each dies, and also at
the end of the year in which A dies. But the present
value of all the rents is a perpetuity diminished by the
value of an annuity on A’s life, or |—|A. And if this
be the value of a fine of 1 . |P. then |—|A, divided by

232 ESSAY ON PROBABILITIES

1+ |P, gives the value of a fine of 1/. in the same cir-
cumstances. Hence the following
Rue. From the value of a perpetuity subtract that

time of renewal.

If there be several lives in the lease, apply the pre- —
ceding rule to each life, and add the results: for the ©
several contingencies do not interfere with or depend —
upon each other, nor will the case of more lives than —
one in one lease differ from that of several leases each on —

one life. The most convenient method is as follows : —

Ruz (for several lives). Multiply the value of a —
sum of the values of the annuities on the different lives: —

perpetuity by the number of the lives, and subtract the

of an annuity on A’s life, and divide the remainder by H
the value of an annuity due on the renewal life at the ©

;

divide the result by the value of an annuity due on the

renewal life at the time of renewal.

Question. An estate of the clear annual value of ©
£a per annum is to be leased on n lives, A, B, C, &c., —
with liberty to renew at the end of each year in which ©
a life drops, the best life in the tables being P: what —
fine should be paid, on the supposition that the pur- —

chaser is to have a given rate of interest for his money?

Rue. Find the value of the perpetuity of £a per :

annum; multiply it by the value of an annuity due on

the renewal life at the time of renewal, and divide by ©

the excess of m times the value of a perpetuity of 12.

over the sum of the values of annuities on the lives of —

A, B, C, &c.: the quotient is the value of each fine

required. But if a sum £8 be paid down, and the rest ©
of the value of the estate is to be paid in fines, then —
subtract s from the perpetuity of £a per annum, before ©

using it in the preceding rule.

Exampite 1. The lives in possession, A, B, and C, — |

are 35, 48, and 60 years of age, and the fine paid on

renewal is 3007. What is the present value of all the 4

fines, using the Carlisle tables, and interest at 4 per
cent. P *

* I have taken Mr, Milne’s example, in order to show the accordance of

ON THE VALUE OF REVERSIONS. 233

| =25 |A=16°041 |P=19°792
3 |B=13°419 I
oa |C= 9°663
75 20°792)35°877(1°7255
39°123 39°123 300
35°877 517°65

In Milne 517°6296

The present value of every single pound of the fine is
1°7255/., which, multiplied by 300, gives 517-651.

Exampre 2. All things remaining as above, the pre-
ceding lease, worth 120/. per annum, is purchased for
25007. and a contract for fines on renewal. What
should the fine be?

|= 25 1+|P=20-792
120 500
3000 35°877) 10896(289°77
2500 289-782 in Milne.
500

The answer is 289-771.

Question. What yearly rental should the fines be
considered as amounting to; and what should be paid
by the lessee annually to an insurance office which would
undertake to pay all the fines as they become due?

These two questions are the same, and the answer to
both is,—the yearly interest upon the present value of
the fines. Thus, in the first preceding example, the
lessor’s interest, at 4 per cent., is worth 20-7/. per
annum for ever; which the lessee might either pay to
his landlord, as a commutation of fines, or to an insur-
ance office, which should take them upon itself.

Question. What is the present value of the next
- fine upon the renewal of the first life which drops of
the three, A, B, C?

the rules. The slight difference arises from Mr. Milne’s rule requiring an
interpolation, which he has very properly thought it not worth while to
make. I have taken more decimal places than those previously given, in
order to show the accordance more clearly. (Milne, page 365.)

QS 4 ESSAY ON PROBABILITIES.

This is evidently the present value of 1/. to be re.
ceived upon the failure of the joint existence of A, B,
and C, and is to be found (page 214.) by subtracting
|ABC, the value of an annuity on the joint lives, from
that of a perpetuity, and dividing by the present value
of a perpetuity due.

Question. If the tenant wish to exchange one life
for another and a better, how much should he pay to be
allowed to do so?

Rue. If the fine be 1/., subtract the value of the
inferior life from that of the better one, and divide the
difference by the value of a perpetuity due on the re-
newal life at the time of renewal. To exchange two
lives, or three lives, use the sums of the values of the
better lives, diminished by that of the inferior ones.

Question.* If the lessor have only a life interest in
the estate on which he grants leases for lives, what is the
value of his interest ?

In strict justice to future holders, it ought not to be
worth more than the rental calculated in the last ques-
tion but three, continued for his life. But it is the
nature of this species of property, that the life interest
of the holder is subject to considerable fluctuations of
value, the preceding annuity being at one time less and
at another greater. A lessor, for instance, who enters
when the lives in possession are very old, himself being
very young, has nearly a certainty of one fine on account
of each, and not much less than an even chance of a
second, while his prospect of a third may be worth
calculating. But a lessor who enters at an advanced
age, against lives which are very young, has a present
interest in each coming fine, which may be determined
oy finding the present value of 1/.,to be paid when the life
drops,on condition the lessor survives(page 223). Indeed,
whatever may be the lives in the lease, provided the
lessor enters at an advanced age, his interest is deter-

* This problem, properly treated, would be of extreme complication, and
J do not remember having seen it proposed. The method in the text is an
approximation,

Cara

ON THE VALUE OF REVERSIONS. 235

mined with sufficient accuracy by finding the values of
insurances on the lives in possession, on condition of the
lessor surviving. But when the lessor is young, I am
not aware of. any rule to which I would trust, as making
as good an approximation to the value of his life interest
as can be made in other cases. Each case must be deter.
mined by its own details ; and it will always be safe to
begin by calculating what we may call the mean value,

namely, the annuity first mentioned ; which may, for

any thing I see to the contrary, be a perfectly correct
mode of proceeding in all cases. |

Question. If the lessor should refuse to renew, and
if it be pretty certain that his successor will adopt the
same course, what is the present value of the tenant’s
interest ?

Evidently the clear annual value of the estate, con-
sidered as an annuity upon the longest of the three lives,
the value of which is determined in page 208.

This awkward contrivance for limiting the rights of
corporators over property is prejudicial in its effects, both
upon the tenants and the lessors. The former, holding
an interest of a comparatively precarious character, have
not the same inducement to improve their property which
is felt by leaseholders for fixed terms of years. On
the other hand, the lessor, in all cases in which he has a
personal interest in the proceeds of the estate, has two
distinct periods of temptation to an act of equivocal
morality. If he be young, he may, as it is called, run
his life against those in possession ; that is, refuse all
renewals, upon the prospect of a large ultimate gain
from the falling in of the old leases: if he be old, he
may induce the tenants, by offering easy terms, to change
their old lives for young ones, thus impoverishing the
successor, by leaving him nothing but long leases, or
leases on young lives.

It must very often be a question for the lessee,
whether it would not be his wisest plan to refuse all
renewal, and to insure a certain sum upon the last sur-
vivor of the three lives by which he now holds. The

236. ESSAY ON PROBABILITIES.

prudence of such a step must depend upon the fine
demanded for a renewal. In the case of church leases,
I believe it could not often be desirable: and certainly
not if they are let so much below their value as has
been asserted. But if the fine demanded should be
exorbitant, it would then become cheaper to insure the
longest of the lives in possession than to pay the demand,
‘The premium for such insurance would be found as in
page 214, the value of an annuity on the longest of the
lives having been previously found in page 208.
Question. The average value* of a living is £s per
annum, and the proper allowance to the incumbent for
the performance of the duties is £v; the unavoidable
expenses at entrance are £f for each new incumbent ; and
the value of the life of the present incumbent is |A,
while that of the new incumbent will be |P. What is
the value of the perpetual advowson of such living?
Rute. The value of a reversion of 1/. per annum
after the death of A being found (by subtracting |A
from the value of a perpetuity), and multiplied by the
excess of s over v, will give the value of the perpetual
advowson, as it would be but for the expenses at entry.

For these, deduct the present value of a fine f, payable

at the end of the year in which each incumbent dies,
the value of each pound of which determined by di-
viding the reversion aforesaid by one year’s purchase
more than |P. The difference is the net present value
of the advowson required. According to a frequent
practice of valuing advowsons, in which the expenses
at entrance are neglected, the buyer pays them twice
over.

* This should include all real profits : for instance, the value of the par-
sonage house as a residence, considered as taken on a strict repairing lease.

Re a RT eS ete he Pe eee RT ee oe

237

CHAPTER XI.

ON THE NATURE OF THE CONTRACT OF INSURANCE, AND
ON THE RISKS OF INSURANCE OFFICES IN GENERAL.

In laying down the following considerations, I think it
right to state most explicitly, that I intend no direct
reflection upon any office now in existence, or whose
establishment is contemplated. In a set of societies so
numerous and varied as those in question, there must be
details in one and another, of which any individual, who
turns his attention to the subject, must disapprove ; but
the studied exclusion of the name of every office what-
soever will, I hope, be taken as earnest of my desire to
confine myself to the enabling other persons to discover
the grounds of censure, without directing their attention
to the quarter in which they are to be found. I have
not much fear that any part of this chapter could have
been misconstrued into allusion ; and perhaps even the
present disclaimer may have no other effect than to make
some imagine that there must be more meaning some-
where than is openly expressed. Leaving such lovers
of mystery to their search, I proceed to the subject be-
fore me.

The avowed levellers in politics, a rare and scanty
sect among educated persons, would have an argument
of some force, from considerations of general expe-
diency, if it could not be shown that any attempt
to equalise property would be attended with a vast di-
minution of the fund itself, so that the great majority *

* The only great alteration Of property which is likely to be agitated, is
the question of the national debt, the entire abolition of which is not with-
out its advocates. This enormous sum, as it appears, is really little more
than one year’s income of the country, and perhaps not so much, if all
the colonies be considered. ‘The honesty of a sponge not being considered,
there would still remain this question :— Would the ultimate loss occasioned

by the subversion of such a debt, amount toa year’s income of the country ?
If so, there would be no gain arising from the abolition uf the debt.

238 ESSAY ON PROBABILITIES.

would really have even less than they now possess, and
also less facilities of increasing their stock. The differ-
ences of talent and of life would still remain, constantly
working towards a restoration of the ancient inequalities,
in which they would almost instantaneously show their
power. A division of property, to be permanent. must
be accompanied by a division of intellect, a division of
manual skill, and a division of life; nor would the sum of
the parts make up the whole in any one of the four, except
the last. A law which should tax the property of all who
live beyond a certain time of life, to provide an addi-
tion to the maintenance of the widows and children of
those who die before it, would not be so utterly im-
practicable, nor so pernicious, as an attempt at equaliza-
tion of fortune, intellect, or skill. Such a law would,
however, fail in its operation, by the mere difficulty of
arranging its enormous details, the frauds to which it
would give rise, and the temptation to idleness which it
would hold out to the young. A small community,
consisting of members of known honesty, living under
a government in which they reposed entire confidence,
and possessing sufficient inducements not to relax in
their exertions, by the certainty of a provision for their
families, might live under such a law: and such com-
munities actually do exist, under the name of insurance
companies,

If a large number of persons, all of the same income
and prospects, and all certain of the same duration of
life, were to choose a common bank in which to de-

posit their savings, each laying by a given proportion

of his income, it is obvious that each would receive the
same sum as the rest at his decease ; but, if the lives
were of unequal and uncertain duration, this result
would no longer be produced. It might, however, be
attained by a covenant, that all sums paid in should
remain till all were dead, and then be equally divided
among the executors of the parties. Such a bank might
be called an equalization office, and it would present

ON THE NATURE OF INSURANCE. 239

the first approximation towards an insurance office such
as those which at present exist.

As yet we have not mentioned the interest of money.
Suppose the equalization office to pay no interest ; and
suppose all the lives to be 20 years of age, such as are
described in the Carlisle tables, the average duration of
which is 414 years. If, then, every person pay 1/. per
annum, each will ultimately receive 414/., which is the
mere compensation of the inequality of life. Such per-
sons would enter into a mutual covenant, by which
those who live beyond the average term would divide
the surplus of their savings among those who fall short
of it.

Probably, if the following question were put to all
those whose lives are now insured, What is the advantage
which you derive from investing your surplus income
in an insurance office? more than half would reply,
The certainty of my executors receiving a sum at my
death, were that to take place to-morrow. ‘This is but
half an answer ; for not only does the office undertake
the equalization of life, as above described, but also the
return of the sums invested, with compound interest.

No one cari form an accurate idea of such an esta-
blishment, who does not tonsider it as a savings bank,
vielding interest, and interest upon interest. This is
the reason why an office which charges for its insurance
more than it is worth, as an insurance, may nevertheless
put its contributors in a better position than they could
have held if there had been no such institution. To
make this clear, let us consider the working of a simple
investment office. A large number of individuals sub-
scribe a sum, which they intrust to an individual or a
company to employ, yielding them the return at some
fixed, but distant, period. Let each share be 1004.
The best thing which an individual could do with such
asmall sum, so as to have perfect security for its return,
would be to invest it in the funds, at 34 per cent. He
might also invest the interest, and thus obtain com-
pound interest: but it is not easy for an individual to

240 ESSAY ON PROBABILITIES.

do this. Unless he provide an agent to draw the divi-
dends immediately on their becoming due, various cir-
cumstances will happen to prevent the immediate in-
vestment of the interest. It is not at all an unfair
calculation to suppose that, upon each half yearly divi.
dend a month will be lost, so that nominal compound
interest for 42 years will only be really for 35 years.
A single pound, therefore, laid up by a man of 20
and improved for the average term of his life, at 34
per cent., would only become 34/. ; while, in the hands
of a person avho lost no time, it would become 41/., or
nearly a pound more. On the other hand, a company,
or a skilful individual who can command large sums of
money, can always make the best interest which the
market will afford. The funds, from the security of
their tenure, and the conveniences which they offer,
will always, in ordinary times, represent the lowest rate
of interest which money will yield. Other investments,
which offer better interest, are generally only accessible
to those who can command considerable sums, and are
frequently attended with risk ; so that it requires know-
ledge to distinguish between the sound and the unsound.
A company, employing the whole time of a person or
persons skilled in money matters, and having continual
large investments to make, can realise not only more
interest, but so much more, that there shall remain a
surplus worth considering, after the skill employed has
been paid for. It is not assuming too much to say
that, all expenses paid, they can command 34 per cent.
compound interest. More than this, they can obtain
such interest without any delay in investing the interest.
The process is extremely simple: it is not difficult to
ascertain what sum should lie permanently at the
banker’s, in order to meet current expenses, so that the
banker has general directions to buy stock as soon as the
balance in his hands exceeds that sum; and all cash
received is paid into the bank at the close of each day.
Suppose it should happen that ten individuals paid 1001.
into the office on account of life insurance premiums, in

ON THE NATURE OF INSURANCE. 244

the same hour in which the executors of a deceased
contributor received a claim of 1002. ‘The hundred
pounds, which, in the theory of the process, should be
sold out, or otherwise set free, to meet the claim, is in
its practice supplied by the new premiums, so that the
premiums of those contributors are making interest from
the hour in which they are paid. But there is always an
unemployed sum lying at the banker’s. This is true; but
the interest of that sum is the salary of an officer of the
institution, namely, the banker himself. All such ex-
penses paid, I believe it may be stated, with correctness,
that an investment office can net 34 per cent. compound
interest. Hence 1/., improved during the average life
of an individual aged 20 years, would become 4'/,

The institution we have hitherto described is simply
an office for the investment of premiums and the equal-
ization of results: it becomes an insurance office when
it undertakes to pay a fixed sum for a fixed premium,
at the end of a given time after the decease of the party.
It then begins to incur a risk of a twofold character:
in the first place, the lives which it undertakes to insure
may not die, one with another, in or near the same
manner as those from which the tables were constructed ;
in the second place, the rate of interest, upon which it
calculates the premiums, may be higher than it is after-
wards able to obtain. According to the Carlisle table,
the premium which should now be paid to insure 100/.
upon the life of an individual aged 20, is one pound
seven shillings, or 1°32/., at four per cent. According
to the Northampton table, and at three per cent., the
same premium should be 2°27. Taking the first pre-
mium, and assuming its table, the office will not be
sure of avoiding loss, until the party has lived 35 years;
by which time the premiums, with their accumulated
interest, will have passed 100/. It is a little more
than 2 to 1 that a life of such an age shall live beyond
32 years after the contract. Taking the premium of
the Northampton table, the party must live 28 years
before the office can gain by him ; and it is about 10 to

R

942 ESSAY ON PROBABILITINS.

7 that he will outlive this term. We have now to
ask, What are the principles which should guide the
office in the determination of its premiums, it being
remembered that there is an absolute security required,
and that the remote chance of bankruptcy, which is
almost essential to the ordinary run of commercial affairs,
is not to be encountered ?

The basis of the tables is the observation of the lives
of a comparatively small number of individuals; it being
well known that the value of life varies considerably in
passing from one class of society to another. Now, we
have seen (page 91.) that we cannot depend upon a
law of probability, derived from a limited number of
instances, with the same degree of confidence, as upon
one which we know to exist d@ priori. If we were sure
beforehand that the great average of life in England
was according to the Northampton, or any other table,
we might rely upon such a document as being extremely
likely to exhibit, with small fluctuations, the future
course of the lives of the two thousand or ten thousand
persons insured in any given office. Let such a table
be assumed, and let the premiums be so calculated, that
it shall be a thousand to one against any ruinous amount
of fluctuation, taking the law of the tables as that which
will certainly prevail in the long run. Then return
from the hypothesis to the truth, and, taking the num-
ber of lives from which the table was actually formed,
say 5000, suppose another 5000 persons to have com-
menced an insurance office. The degree of fluctuation
within which if was 1000 to 1 that the future re.
sults should be contained, is now larger than before,
in the proportion of the square root of 2 to 1, or
in that of 14 to 10, nearly. Larger premiums would
then be required to make ruinous fluctuation as unlikely
as upon the preceding supposition. ‘These considerations,
which may easily be reduced to calculation by the rules
in chapters IV. and V., will serve to show that there
may be danger in the assumption of any table formed
from experience: and they ought to operate powerfully

ON THE NATURE OF INSURANC2. 243

as a caution against lightly admitting a change of pre-
miums, on the authority of any small number of facts.
But more particularly should they be attended to in the
formation of new varieties of contingency offices, the
chances of which have not yet stood the test of ex-
perience.

But there are reasons why the premiums of an in-
surance office need not be so high as the very limited
number of data in their tables might seem to require.
If the fluctuations from the average, which are within
the most cautious definition of reasonable probability,
were all to be encountered at once, or might be en-
countered at once, it is difficult to say what premiums
should be considered as too high. But this cannot be
the case, unless, indeed, a pestilence should single out
tiie members of an insurance office, or an earthquake
should, by one extraordinary event, swallow them all up
in the place where, by a most remarkable coincidence,
they were all assembled together. Such extreme cases
are not worth consideration; and we may take the
chances of life and death as distributed over a large
number of years. In the meanwhile the surplus fund
increases at compound interest; and the problem is, not
whether a given number of lives will, on the whole,
drop so much before the predicted time that a given
fund will be destroyed, but whether this can happen so
fast, that it will outrun the increase of the fund at
compound interest. If, indeed, there were compound
mortality to set against compound interest ; that is, if
the number of deaths must become larger from year to
year, or if the rate of mortality were increasing, the
fear of such a result might be entertained ; bué all
experience is on the other side, and tends to show that
the value of life is increasing, instead of decreasing.

The tables of an insurance office must be considered
as collections of limited data, the premiums deduced
from which are increased by a percentage, to meet the
possible fluctuations of mortality. As soon as these
tables are formed, and the directors have published

R 2

24.4, ESSAY ON PROBABILITIES.

their proposals, an insurance office is created, with all
these fundamental characters already described, and
which are but ill represented by the term. The word
insurance or assurance has given rise to some wrong
notions, and it will be worth while to examine the
nature of the contract.

A and Co. engage with B that, in consideration of
17. a year, paid by him during his life, they will pay
20/1. to his representatives as soon as he shall be dead.
Both parties run a risk; A and Co. that of having to
pay B more than they receive ; B, that of paying more
than will at his death produce 207. But the risk of
the office is of immediate loss, and that of B, of deferred
loss: that of the former is also continually lessening, and
that of the latter increasing ; until, should B live long
enough, both risks become certainties. If the in-
surance be only for a term of years, B runs the risk of
losing his premiums altogether.

The office does not inquire what reason B may have
for insuring his own life or that of another person, nor
do any possible contingencies, except those of life, affect
the office calculafions. We cannot, therefore, be too
much surprised at the ignorance shown by that judge*
who declared that life insurance ¢ was of its own nature
a contract of indemnity ; that is to say, if, by any
lucky chance, B can be proved to have accomplished
the object for which he insured by other means, he has
no claim upon the office. The circumstances are as
follows ; and the absurd conclusion is law, and

would be practice, if the insurance offices had not re~

fused to acknowledge the decision, or protect themselves

by the precedent. A and Co. covenanted with B to —

pay 500/., if C should die within the term of seven
years next ensuing, in consideration of the usual pre-
mium. C did die within the term ; and A and Co., in

* Godsall' v. Boldero. See the report of the case in Mr, Babbage’s
* Comparitive View of Institutions for the Assurance of Lives.”

+ He might have said that the law would refuse to consider an assurance
in any other light ; but he was palpably wrong in asserting that the con-
tract, as understood by the parties, was merely one of indemnity.

ON THE NATURE OF INSURANOR. IAS

answer to a claim of 500/., replied, that the intention
of B in insuring the life of C, was to obtain security
for the payment of a debt of 500/., due by C to B,
which debt had been already paid by C’s executors:
consequently they owed nothing to B. An action was
brought by B, and defended by A and Co. on the above
plea; and a special case being made, the point was de-
cided by the court of King’s Bench against the plain-
tiffs; thereby establishing the principle, that life in-
surance is a thing similar to fire or ship insurance ;
namely, a contract of indemnity, to be fulfilled with
allowance for salvage.

The defendants’ case rested upon the asserted nature of
the contract, and the statute 14 Geo. III. c. 48., which
enacts, that “no greater sum shall be recovered from
the insurers than the amount or value of the interest
of the insured in such life.” The act does not state
at what time this interest is to be reckoned, but the
plaintiffs contended that the time of death was the
meaning of the statute; the defendants averred, and
the court decided, that the time of bringing the action
was to be understood. The plaintiffs contended that
the debt was not the object of insurance, but the life
of theinsured ; the court decided, that ‘* This action is,
in point of law, founded upon a supposed damnification
of the plaintiffs, occasioned by the death, existing and
continuing to exist at the time of the action brought ;
and, being so founded, it follows, of course, that if,
before the action was brought, the damage which was
at first supposed likely to result to the creditor were
wholly obviated and prevented by the payment of his
debt, the foundation of any action on his part, on the
ground of such insurance, fails.” This sentence con-
tains nothing but very good sense, and, no doubt, very
good law: but the application of it was accompanied
by a mistake as to the nature of the damnification
which the plaintiffs had sustained. The counsel on
both sides, the court, the insurance office, and the
plaintiffs themselves, showed a very partial knowledge

R 3

294.6 ESSAY ON PROBABILITIES.

of the nature of the contract ; and I make no doubt,
that almost every person who heard it agreed with the
court, however much they might impugn the decision
on other grounds, that the damage* to the creditor
“* was wholly obviated and prevented by the payment
of his debt.”

In order to show that such was not the case, we
must suppose that an exactly similar transaction had
taken place before any insurance office existed. How
this could have been may not be apparent, if we
take the notion which the law formerly entertained of
such an office ; namely, that it is a species of gambling
house: but if we prefer to consider it as a savings
bank, with an equalization system (page 238.), which is
unquestionably the correct notion, we may return to the
circumstances which the case would have presented had
there been no insurance. C, a person whose credit has
become doubtful, is indebted to B to an amount which B
could not afford to lose ; consequently, B, knowing that
his chance of payment is precarious, resolves to diminish
his expenses, hoping by economy to restore to his family
the sum which he may have lost by his engagements with
C. He collects, accordingly, a small fund, which he
places with his banker, avowing the purpose of its
collection. In the mean time C dies, and some friends
pay off his debts, and that due to B among the rest.
The latter having now no further occasion for such
economy, draws upon his banker for the amount, and
is answered, that, since the purpose of the saving was
fulfilled by the payment of C’s debt, he, B, has no
further claim upon his own money. An action is
brought, and the courts decide that the banker is right,
and that B, having really attained his object in one
way, has no right of property in the proceeds of another
attempt to serve the same purpose.

* The defendants paid into court a sum somewhat less than the amount
of the premiums they had received from the plaintiffs, doubtless as a pre-
caution, in case the court or jury should think the premiums ought to
have been returned.

ON THE NATURE OF INSURANCE. 247

The only distinction between the case just put and
that which actually occurred is, that the banker was a
person who gained his profits by receiving such savings
during a contingent term, and guaranteeing a tixed
sum ; standing the loss, if there were any, and paying
himself for it out of the gain which would accrue in
another instance: the premium having been calculated
so as to insure a moral certainty of profit upon the
average of similar cases. It is not pretended, on either
side, that the chance of indemnification at the hands of
C’s executors was made to lessen the consideration paid
by B for the guarantee; and the legal iniquity of the
decision may, I think, be made clear, as follows :—

It will hardly be disputed, firstly, that the legislature
is the judge of what shall constitute valuable consider-
ation ; and, secondly, that a consideration which is ex-
pressly allowed to be good in a statute, should be ad-
mitted as such in the decisions of the courts. Now,
the contract of insurance, be it gambling, or be it not,
rests entirely upon the permission given by the law to
consider a high chance of a small sum as good con-
sideration for a low chance of a large sum. If I now
pay 2/. of premium for 100/., in case I should die in
a year, and if my executors can maintain an action for
100/., it must be because the law sanctions the notion
that 2/., nearly certain, may, with consent of parties, be
considered as an actual equivalent for a distant chance
of 100/., as much so as one weight of silver for another
of bread, or food, clothing, and wages for personal
service. It is true that the same law, fearing certain
reputed immoral practices, to which the power of
making a particular bargain offers temptations, may
limit the circumstances under which it will permit
such bargains to be made ; but this is equally true in
regard to the other sort of contracts mentioned:
indeed, there is no sort of bargain which is not
under regulation. The law, then, allows risks, and
permits unequal chances to be compensated by giving
odds ; the courts declare that, after the cast shall have

R 4

248 ESSAY ON ‘PROBABILITIES.

been made, and one of the parties shall have stood his
risk, which turns out in his favour, the other party
shall receive an ex post facto release from the condi-
tions of his bargain, because circumstances afterwards
arise, which, had they existed* at the time of making
the bargain, would have made it illegal. The several
principles on which the decision was founded, well
carried out, as they say in parliament, would require
that the previous contracts of a man who becomes
insane should be null and void ; that the meat which
a man buys for his dinner should be returnable to the
butcher under the costt, if a friend should invite him
in the mean time ; and, in the case before us, supposing
that C should have outlived the term, and his debt
were paid, as before, then B might have brought his
action against the office, for the return of the premiums;
alleging that, as it turned out, the office would have been
indemnified, and, therefore, should be considered as
having run no risk. |

But, said the judge, the damage was “wholly” ~
obviated and prevented by the payment of the debt. —
To try this point, let us make a debtor and creditor ~
account of the whole transaction. The following is
the way in which it will stand.

Cr. Dr
£500 worth of goods fur- £50U paid by C’s executors.
nished to C.
Certain small premiums Those same premiums re-

paid to an insurance office, turned by the office, instead
with imminent risk of their of £500. mo
entire loss; such premiums,

multiplied by the risk of loss,

as in chapter V., being good

legal consideration for a re-

mote chance of gaining £500,

and so considered by both

parties.

* This is admitting more than is absolutely necessary ; for, unless there
were mathematical certainty that a third party would step in and pay C’s
debts, it is difficult to see how B’s insurable interest would cease.

+ The sum paid into court by the insurance office, was less than the
amount of the premiums: but the plaintiffs waived that point.

ON THE NATURE OF INSURANCE. 249

The advantage of the moral security which a contract
of insurance gives is obvious in the transaction which
led to this decision ; namely, the insurance of the life of
a creditor by a debtor at his own expense. Commercially
speaking, such a transaction is literally this: C owes
500/. to B, who, doubting his chance of payment if the
debtor should die, buys 500/. from a third person, and
’ makes believe that it is the 500/. which C owes him.
Morally speaking, it is the determination of B to retrench
his own expenditure, as soon as he finds that a part of
his property consists in bad debts. This the office
enables him to do in a manner which will make the
retrenchment proportional to the necessity for it. In
the mean time, it is much to be wished that the law of
life insurance were settled upon a fixed basis, which
should proceed upon such a definition of the contract as
has been here explained, and not on the notions which
have been drawn from a supposed analogy between it
and the insurance of a ship or a house. The effect of
the present state of the law is, that the offices have no
law except that of honour, which, though it more than
suffices for the protection of the insured, yet may at
any time involve the offices in the necessity of paying
really questionable policies, without having the means of
submitting to open examination the point on which
they wish to resist. Policies of insurance are sold daily
to persons who have no interest in the lives of the
insured parties, on the faith of the good conduct of the
offices. If an office were to resist the payment of a
policy so transferred, say on the ground of fraudulent
representation, the parties so resisted might give out
that the opposition of the office arose out of an intention
to cover themselves by the present letter of the law.
Neither could such a case be carried into court without
proof that the plaintiffs possessed that insurable interest
in the life of the deceased which the law requires.

The nature of the contract, both in law and usage,
having been laid down, we must next ask what are the
means which the offices employ to reduce the risk so as

250 ESSAY ON FROBABILITIES.

to render themselves safe against fluctuation. The state
of opinion upon this matter is somewhat unsettled ; one
party advocating the practice of approaching near to the
line which separates security from insecurity ; another
insisting upon what appears to the first a most super-
fiuous degree of caution. Without expressing an opinion,
I will describe the various risks, and the method of
avoiding them which has usually prevailed.

1. The insecurity of data, that is, of existing tables
of mortality. This divides itself into two parts ; that
relating to the young and middle aged, and that relating
to old lives. With regard to the first, the data might
probably be obtained in sufficient numbers to justify a
considerable degree of confidence in them as to the
chances of a single life, or even of a considerable number ;
but when the number of lives is to be as great as the
number of persons who may choose to offer themselves, the
considerations in Chapter IV., again adverted to in page

242, present themselves in force. J am not aware that -

any writer on the subject in this country has formally
taken into consideration the uncertainty of tables, arising
from their limited numbers, except Mr. Lubbock, who
has made use of (Cambridge Phil. Trans., and Treatise
on Probability Lib. Usef. Know.) the correction which
the probability of living a given number of years should
receive on that account. But, considering the probable
errors of the data, this correction is small, and the
question how far an office proceeding upon such data
can deal with the public to any amount is yet in its
infancy, though the necessity for its consideration is
approaching, and it is one of vital importance to the
interests of the middle and lower classes.

The constructor of tables of mortality draws a number
of balls from an urn which contains an infinite number
and, having sorted them into red, blue, black, &c.,
presents them to the world as a necessary representation
(or very nearly so) of the proportions in which those
colours are scattered throughout the whole urn. He
commits an error which is in all probability very small,

$$

ON THE NATURE OF INSURANCE. Zot

and which has hitherto been carefully guarded in the
deduction of office results. But there is a much more
important question behind. Suppose the calculator had
undoubtedly succeeded in exhibiting the real law of
mortality, and that it were quite certain the next
hundred million inhabitants of Great Britain would die
in the manner pointed out in a table. In such a
case, many will say, the office may charge the real
premiums deduced from the table, with a very slight
addition for expenses of management. They may leave
the fluctuations to take care of themselves, and trust in
the long run. This assertion I now proceed to discuss.

If the banker of a gaming-table were to follow the
same plan, that is, if he were to stake against all comers
with only just enough of advantage to cover expenses,
he would infallibly be ruined at last. It might not be
in this year nor the next, nor in this century nor the
next; but ruined at some time or other he must be
(see page 110, and also Appendix J.). If the case
of the office managers were precisely analogous to that
of the bankers of the gaming-table, I would repeat
with as much confidence of the former what I have said
of the latter. But, in the first place, the fluctuations of
mortality are not, by very much, so great as those which
take place in the assortment of cards, nor even so great
as those which take place in harvests, in the price of
provisions, &c. This is much in favour of the insurance
office ; but who can say how much?

In the second place, the fluctuations of mortality
have of themselves a tendency to create opposite fluc-
tuations. Thus, a very sickly season carries off the weak,
and deprives the succeeding years of those who were
most likely to have died; causing, therefore, a season of
remarkable health. This is a very important item in
the theory of the fluctuations of mortality, and there is
“hothing similar to it in the case of the gaming house.
It reduces annual fluctuation itself to a species of regu-
larity, and is, perhaps, the sufficient reason for the slight-
hess of the total fluctuations.

252 ESSAY ON PROBABILITIES.

In the third place, with a merchant or a banker, the —
liability to a demand and: the demand itself come so ©
nearly upon one another, that real insolvency and bank- —
ruptcy are never far asunder. When credit cannot be —

sustained by monthly, and even daily, proofs of substance.

it takes its departure altogether: but it is not negessarily
so with an insurance office, of whose existence it is the —
essence to be always receiving consideration for bills —
which, one with another, have a long time to run, —
Such an establishment, as will presently more distinctly ©
appear, may be in reality insolvent many years before —
the symptoms of bankruptcy come on. As no large

concern of the kind has hitherto failed, it is difficult to

say how they would finally come on: but this much is —
certain, that an insurance office which could really pay —
only ten shillings in the pound might, by introducing a |

better system, or by mere force of circumstances, not
only recover its ground, but ultimately become exceed-
ingly profitable. But I throw this part of the argument

(though it shows a strong principle of vitality inherent —

in the constitution of such offices) out of the question ;
for, surely, no sane and honest person would trifle with

bes

important matters so far as to assert that the possibility

of temporary insolvency, to be redeemed by the chapter —

of accidents, or prudence, when it was wanted, should

enter into the deliberate calculations on which men
should be invited to stake the subsistence of their —

children.

If the last contingency be rejected, that is, if it be —
held absolutely necessary to calculate on permanent ~
solvency, both real and apparent, then I assert that —
there is not sufficient ground to gainsay the conclusion, -
that any insurance office charging only vea/* premiums
(increased for expenses of management) must inevitably —
have its phases of solvency and insolvency, at the very ©

best. Begin by considering the office as identical in

principles with the gaming house, and beat down the

* By real premiums I mean those which only cover the risks of life.

Be ig Sad 5 Be eters $
AC eae ene ia Ne

Doe

ON THE NATURE OF INSURANCE. 253

certainty of ruin which is thus known to exist, if they
play upon equality of chance, by allowing for the first
two of the three preceding considerations. There must
still remain more risk than it is safe to face of insolvency,
either temporary or permanent. And though, in con-
sequence of the smallness of the portion which the office
risks upon one hazard, a very small mathematical ad-
vantage might be sufficient, yet, so long as the necessity
- for such an advantage exists, and its absolute amount is
unknown, so long must an office guard itself by requiring,
in the first instance, a sensible addition to the real
premiums.

With regard to the old lives, there is an additional
ground of insecurity. Not only are the probable errors
of the tables exceedingly large with respect to them,
but, from the smallness of the number which will
enter an office, there will be a liability to great fluc.
tuations in the results of transactions with them. The
first circumstance would prevent the second from be-
coming ruinous, but at a risk of loss to the capital
invested by younger lives: it is usual, therefore, to
exclude all lives above a certain age from entering the
office, upon the principle that no risks are to be taken
of which the numerical amount is not well understood,
and of which the number is not large enough to secure
an average. But, since the tables of old lives are only
a very unsatisfactory approximation, and since the pre.
miums payable by young lives depend in part on the
chances of those lives becoming very old, how does it
happen that the insecurity of the latter part of the
tables does not affect the premiums throughout? It
does affect them, but not sensibly, for the following
reason. If, assuming the Northampton table, we sup-
pose a person aged 40 to insure his life, we see that
the portion of the present value of his insurance which
depends upon his dying in his 85th year is very small,
on two accounts: firstly, because the chance of his
living to the age of 84 is very small; and, secondly,
because the present value of a sum to be received

254 ESSAY ON PROBABILITIES.

45 years hence is small, compared with that of a sum
now due, or receivable soon. This last consideration
works as follows: — When a percentage comes to be added
to the whole present value or to the premium deducible
from it, for the security of the office, that percentage
being made upon a much larger sum than the present
value just mentioned, a very trifling deduction from
the whole additional sum will cover a very serious
mistake in the mortality of the older years. For ex-
ample, in the Northampton tables, the chance of
40 living to 85 is about ;',, and the present value of
1/. due in 45 years is about 5s. at 3 per cent. From
this it follows that 100/., to be paid if a person aged 40
dies after 85, cannot be worth so much as 1°25/. But
the present value of the whole insurance is 53°8/. ; and
if this be the real value, and 10 per cent. be added for _
security, then 5°38/. is added; so that if 1°25/. were —
considered as added solely for the chances after 85 years,
it follows that we might consider ourselves as having*
allowed for not being able to calculate the chances on
old lives within one half, and as having added 8 per
cent. to the whole present value besides. Thus, it ap-
pears that our comparatively little knowledge of old
life, though not unimportant, yet can be made to be of
less importance than might have been expected by
one who has not considered the matter. Of course, —
the preceding reasoning must be considered only as _
addressed to a person to whom, for any thing he sees
to the contrary, it is of as much consequence to know
the entire law of mortality in the insurance of young
lives as of old ones.

There is one use of the table of old lives, by which
an insurance office might make its existence very pro-
blematical, to use a gentle term ; namely, by inverting
the order of security, and selling guaranteed* benefits,
which are to increase with the age of the party, and to.
be accumulated solely out of his premiums. To take

* This, of course, does not apply to divisions of profit gained, but to
contracts for sums to be accumulated after the date of the engagement.

ON THE NATURE OF INSURANCE. 255

an extreme case, suppose an office should name a pre-
mium for which it would undertake to pay 100/., if the
party dies in the subsequent year ; 200/, if he dies in
the second subsequent year; 300/. if he dies in the
third ; and so on. In this case, every fluctuation which
bears the appearance of lengthened life, were it only to
amount to deferring one death for a single year, would
be a new claim of 100/. upon the office. The fluc.
tuations which are observable in the very old lives, would
become matters of extreme importance; and though,
assuming a given table fairly to represent the average,
premiums might be calculated which should be sufficient
in the long run, yet there is no possibility of saying
what capital might become necessary to meet the fluc-
tuations of half a century. Such an attempt as the
preceding can be compared to nothing but gambling,
and its stability to nothing but that of a ship running
before the wind, with all the heavy cargo lashed to the
topgallant mast. Other cases might be mentioned,
which should partake of the same species of danger in
a less degree ; but every attempt to guarantee increased
benefits with increasing life should be looked at with
caution, as being of its own nature the addition of
risks in which the errors of the unsafe part of the
tables are, or may be, multiplied into importance.
There is an opposite plan, which I am not aware has
been tried, but which I should strongly recommend to
any new insurance office, as being of a safe character,
and also meeting the views under which many insure
their lives. It is that of insuring decreasing sums,
upon either fixed or decreasing premiums. Many
persons are so situated, that they will be able to provide
for their families if they live a few years. To provide
for the hazardous period, they are under the necessity
either of insuring for their whole lives, that is, of buying
- More insurance than they want; or of insuring for a
fixed term of years, which does not meet several con-
tingencies ; or of making complicated survivorship in-
Surances. But, if a person so circumstanced found, by

256 ESSAY ON PROBABILITIES.

estimation of his income, that he should want 5000/. if
he died in one year, 4800/. if he died in the second,
and so on, it would be desirable that he should be able
to insure for these several sums, contingently upon his
dying in any of the several years to which they are
made to belong. Various modifications of this scheme
might be proposed, all having this difference from the
usual plans, in that the latter enable a person to make
a provision for his family, while the former would only
supply the deficiencies which his death would leave in
the proceeds derived from other sources. In an ap-
pendix (on the value of increasing annuities) will be

found the method of calculating the present values of —

such insurances.

2. The possible fluctuations of the rate of interest.
These may be either general and national fluctuations,
or alterations in the value of the property held by the
office. The former cannot be guarded against or pre-

dicted ; and, as the rate of interest has been slowly —

falling for centuries, there is some reason to suppose —

that this depreciation of money may continue. But '

this gradual sinking of the rate of interest may be only ©
partly dependent upon the fall of profits, and part may ~

be due to the increase of security. 1 question whether ~
the political economist has found the historical materials
for determining this most important element ; namely, —
the extremes of interest at which loans were contracted —

eee er as

in the different periods of our history. The legal ©
maximum of interest, at the beginning of the reign of ©
James I., was 10 per cent., and at the end of the cen- —
tury, 6 per cent. But, at the beginning of the century, —
land was commonly bought at 20 years’ purchase, and —

never at less than 16 years’ purchase ; while at the end —

of the century it was still at 20 years’ purchase. No
method of proving such a point is better than the —

examination of the works on interest which appeared —

during the century. If, then, we suppose, with Adam ~

Smith, and I believe with most others, that the changes —

in the legal maximum of interest followed, and did not

ON THE NATURE OF INSURANCE, 257

precede, those of the market, there is good ground for
imagining that the diminution of the rate of interest
between borrower and lender (from 10 to 4 per cent.)
has arisen more from the increase of security than
from any other cause. If such be the case, there is
strong presumption that the fall is near its end. But,
if the preceding surmise should not be well founded,
and if (as was the case in Holland during a part of the
last century) the rate of interest should fall until go-
vernment can borrow at 2 per cent., and others at 3 per
cent., the change may happen in a manner which will
seriously affect the insurance offices, unless it should
come about so gradually that they will have time to
introduce new premiums for incomers, and surplus to

meet the claims of those to whom they are already

engaged. It is, in the meanwhile, a question well
worth the attention of those connected with them, what
the causes have been which have determined the rate of
interest, and the rapidity and amount of its variations.

The offices depend for the existence of their present
system upon the national debt ; and they are differently
situated from the government which owes the debt, in
that the engagements of the latter are all maxima,
while theirs are minima. If the rate of interest should
really fall, the government will. have the means of
reducing the interest of the debt, never to rise again ;
while the offices have, in fact, guaranteed to their ex-
isting customers a rate per cent., which is never to fall
during their lives. The rate assumed by the offices
should, therefore, never be above that at which the
government can borrow.

With respect to the second reason for a variation in
the rate of interest, as experienced by the office, namely,
a depreciation in its own property, such an establish-
ment, not being allowed to run the usual risks of
mercantile life, should not deal in any but the most
secure investments, and those which depend on the
personal security of others should be altogether avoided.
The only point which it is incumbent to mention, in

s

258 ESSAY ON PROBABILITIES.

addition to general cautions, is a mistake to which such
offices are subject in the valuation of their property ;
namely, the estimation of different items by their re.
puted worth, or by the price which was given for them,
instead of the actual income which they produce. We
shall see the effect of such a mistake in considering
the proper method of inquiring into the state of their
affairs.

The precedent are the contingent risks to which an
office is subject: its certain expenses are the ordinary
charges of management, including rent, salaries, interest
of sums lying at the banker’s, &c., advertisements, and
the commission, as it is called, which most of the offices
pay to those who bring them business.

Commission, in general, means either a per centage
paid to a factor for the transaction of business, or avolun-
tary relinquishment in favour of the person who brings
business of a part of the profit which the said person,
being honourably free to choose between one competitor
and another, has brought to the trader who, there-
fore, allows the commission. It answers to the profit
which the retail dealer is allowed by the wholesale mer-
chant from whom he buys. But, when an insurance
office announces to the solicitor, attorney, or agent of a
party desiring to insure, that they will allow him a
liberal commission, the term has a different meaning.
As between one office and another, the attorney is in a
judicial capacity ; and, as regards his client, he is
already the paid protector of the interests of another per-
son. Hehas, therefore, no liberty of choice between one
office and another, but is already bound to choose that
which he judges best for his client. All who have
written on the subject of late years have attacked this
bribe, for such it is ; but they have directed all their cen-
sures against the offices, as if they were the only parties
to blame. If, indeed, the bribe had been offered to the
needy and ignorant only, this partial distribution of
blame might have been allowed ; but when the parties
who receive the bribe are men of education, and moving

ON THE NATURE OF INSURANCE. 259

in those professions which bring the successful to afflu-

ence, I do not see the justice of allowing them to escape.

I have little doubt that an increasing sense of right and

wrong will banish this unworthy practice, either by

failure of givers or receivers. A barrister cannot offer

an attorney commission on the briefs which he brings,

nor can a physician pay an apothecary for his recom-

mendation ; a jury never receives a hint that the plaintiff
will give commission on the damages which they award ;

and the time will come when the offer of money to a

person whose unbiassed opinion is already the property

of another, will be deemed to be what it really is,

namely, bribery and corruption. It is one among many*
proofs how low is the standard of collective morality ;

and how easy it is for honourable individuals to do in

concert that from which they would separately shrink.

It appears, then, from all which precedes, that the
ordinary risks of an insurance office are alterations of,
and mistakes in determining, the rate of mortality, and
reduction of the rate of interest: which are guarded
against by assuming a rate of mortality beyond all
question greater than exists, and a rate of interest below
that which the funds will yield. At the peace of 1815,
every insurance office used the Northampton table at
3 per cent. This was at a time when the real rate of
interest was higher than at present, and the offices must
have made considerable profit. It was well known that
they did so ; and, accordingly, new offices were formed,
and have continued to be formed up to the present
time, some upon lower premiums than others, and most
of them returning all or part of the profits to the in-
sured. At the same time, an opinion has become very
prevalent, that it is possible for such offices to maintain
their ground at much lower rates of premium than
those in use ; a notion which I proceed to examine.

Mr. Finlaison, whose experience in such matters is
well known to the public, and for whose opinion I en-
tertain a high respect, stands foremost among those
who contend for low rates of premium, having pub-

S 2

260 ESSAY ON PROBABILITIES.

lished a table, which he certifies to be “ abundantly safe
and practicable,” and “ so high as to insure, beyond all
doubt, a surplus of profit ;” which table charges pre-
miums at the ages in which most insurances are made,

falling short of those actually in use about 15 per cent. —

These premiums are supposed not chargeable with the
management of the office, and at a rate of interest of
34 per cent. I take Mr. Finlaison’s proposition as a
modified one, for there are some which go beyond it.

On the other hand, the late Mr. Morgan could never
be persuaded that it was safe to abandon the North-
ampton table ; and considered that the superior vitality
of the members of the Equitable was altogether a con-
sequence of their being select lives. He seems to have
thought that, whatever run of success an office might
have, it should always be on the look out for reverses ;
and that even the enormous accumulations of his office
were no more than the seven good harvests, a provision
for other seven of a different character.

In holding an opinion which comes between that of
these two authorities, I form it on a ground on which
neither would have rested the truth or falsehood of his
own. I consider the fluctuations of mortality as very
little to be feared, compared with those of the rate of
interest. It has long been matter of observation, that

the phenomena of the natural state of man vary but —

little compared with those of his social condition. The
price of provisions swings to and fro like a pendulum ;
the variations of mortality which follow its changes,

though sensible, bear no proportion to the magnitude of —

their cause. The rate of interest has been halved
within the memory of man, and a heavy war might

double it again. That same war, with all its casualties, —

direct and indirect, included, would not alter the mor-
tality of the country by any serious amount. I con-
sider it, then, as next to certain, that the insurance
offices have more to look for, whether as matter of hope
or fear, from the fluctuations of the rate of interest,
than from those of mortality. If the interest of money

" 2 atte Waa piiiin ane
as ee ee PEN eae SR te aerate
SE a ae ne as Sg glia i is 7 ox

ON THE NATURE OF INSURANCE. 961

could be made as stable as the duration of human life,
I could then see no objection to an immediate and con-
siderable reduction of the premiums charged, to an
amount at least equal to that proposed by Mr. Finlai-
son. But here lies the difficulty; that these tables, at
31 per cent., already involve a rate of interest which
the office cannot much exceed, if at all ; so that the secu-
rity which the precautions, nominally made against mor-
tality, really afforded against fluctuations of interest, is
partially or wholly destroyed, while no safeguard is in-
troduced to supply its place.

An office raises its premiums either because its pre-
vious notions of existing mortality were wrong, or be-
cause it finds that it had calculated upon too high a
rate of interest. A mistake on either of these points
might be compensated by a contrary mistake as to the
other. Now, though the offices which existed during
the war have demonstrated that the mortality and rate
of interest together yielded a large profit, it by no
means follows that one of those causes of profit may be
fully corrected, while the other has been correcting
itself. To make both perfectly accurate, would bring
the office to the very line which divides security from
insecurity ; a position which it would not be safe to en-
deavour to maintain. We are already in a very dif-
ferent position as to the rate of interest, which has been
gradually falling since the war. The opinion as to the
extent to which tables of mortality may be safely cor-
rected, is formed upon arguments which dwell on the
favourable rate of mortality, without sufficiently consi-
dering the counterpoise (for, as far as it goes, it is a
counterpvuise) existing in the alteration of the value of
money.

Assuming the necessity of calculating upon a rate of
interest something less than that which can actually be

- attained, I should think that no office would be justi-

fied in supposing more than 3 per cent., with tables

which are sufficiently high to come any ways near to the

actual experience of mortality. With regard to one
8 3

262 ESSAY ON PROBABILITIES.

point, and that of fundamental importance, namely, the
possibility of a still further fall in the rate of interest,
it may even be doubted whether, with such tables, a still
lower rate of interest should not be allowed. But I
am not here advocating one result or another, but only
the necessity of taking into consideration all the pos-
‘ sible sources of danger. To those who would use tables
of greater vitality, I concede that, so far as mortality
alone is concerned, the alteration is admissible ; and for
this reason, that experience shows human life to be
of a higher value than formerly ; but the concession
is accompanied by the requisition of a lower rate of
interest, and for the selfsame reason, that experience
shows the value of money to be less than it was.

The preceding conclusion is reinforced by the con-
sideration, that the worst is to be made of every cir-
cumstance in our, previous calculations. When mor-
tality is diminishing, the whole diminution is not to be
allowed ; but when it is increasing, a larger increase is
to be contemplated. A person who would walk dry-
shod on the sea shore, must not advance so fast as the
ebb, and must retreat faster than the flow. Upon this
consideration, the necessity of providing for a further
fall in the interest of money is increased ; or, which
amounts to the same thing, the amount by which the
favourable alteration in the rate of mortality may be
allowed to affect the premiums is less than it would be
if it were certain that the value of money would remain
unaltered.

A very common security or guarantee to the public
is the announcement of a large subscribed capital, either
paid up in whole or part, or liable to be called for.
This is equivalent to the personal security of a number of
shareholders, collectively making themselves answerable
for the engagements of the office up to a certain amount.
Such a provision in itself is an obvious good ; but, it being
remembered that this security must be paid for, it be-
comes a question how much it is worth, and whether it
may not be bought at too high a price. It is easily

ON THE NATURE OF INSURANCE. 263

understood that the consideration which tempts men to
lend names or money to an insurance office, is the offer
of payment for the risk, or of higher than market interest
for the money. If the capital be paid up, the office
makes common interest upon it, which is returned, with
an augmentation, to the proprietors: if the capital be
only paid in part, or merely nominal, still the office has
to pay something more than it receives.

Now, I take it for granted that an office charging
premiums* such as are commonly demanded, managed
with prudence and economy, and successful in obtaining
business, will not ultimately need any capital at all:
firstly, because the premiums are such as must, in the
long run, realise a profit after paying the expenses of
management ; so that the only use of the capital would
be as a provision against extraordinary temporary fluc-
tuation: secondly, because a sufficient supply of business
renders the probability of ruinous fluctuation extremely
small, and altogether beneath consideration.; Now,
since it is well known that the premiums are sufficient,
it follows that the only need which a commencing
insurance has of capital is for safeguard against the early
expenses of management, and against failure of business;
as follows,

The expenses of carrying on an insurance office,
though they vary somewhat with the amount of business,
yet do not by any means increase as fast. In the first
year of its existence, it would not be surprising if all
the premiums paid were swallowed up by house-rent,
salaries, &c.; while, in process of time, increase of
business might reduce such expenditure to 2 per cent.
upon the yearly premiums. Some capital, therefore, is
necessary at the commencement ; for, if there be none,

* If the premiums were really too low, capital would be an injury, and
not a benefit ; for, since this capital is really paid for, in whole or in part,
out of premiums, it would noc preserve the office from insolvency, but
would rather accelerate its progress towards bankruptcy.

+ The most probable cause of ruin to the insurance offices, or rather the
least improbable, is a national bankruptcy. Any contingency, then, which
is much less likely than a national bankruptcy, need not be considered.

s 4

264 ESSAY ON PROBABILITIES.

those who first insure their lives are entirely dependent
upon the future success of the office. But this capital
need not be large: in the present state of things, an
engaged capital of one hundred thousand pounds is
certainly above the mark, even for an office which is
entirely without connection, and starts without one
single life insured. If, as very often happens, a tolerably
large number of customers has been obtained before the
prospectus of the office is announced, then a capital, the
interest of which will cover the expenses of management,
is sufficient. But here it must be observed that the
proprietors of this capital run some risk of losing a
portion of their principal, and a still greater one of
losing the interest for a limited time. This risk is the
greater the smaller the original subscription, and it
must be paid for accordingly. At the same time, it must
be remembered that the mere existence of the capital
diminishes the risk, by making it the interest of every
proprietor to procure business for the office. The con-
nection thus created is the secret of the successful start
which has frequently been made; and it may be con-
sidered as very unlikely that an office will fail, from
want of business, which is so well supported in the first
instance as is implied when a capital of the preceding
amount is announced.

There is, however, one case in which a larger capital
is desirable, and even requisite ; that is, where an office
is established which is to insure some new and yet
untried risk. Whatever pains may be taken in such a
case to procure facts and deduce proper tables, there is
always a risk that the experience of the office may be at
variance with the facts of the tables. When, for instance,
the general conclusions drawn from the mortality of
towns were first applied to the insurance of life, it was
a risk of unknown amount as to whether the lives of
those who would come to insure would be of the same
class as those from which the tables were made. They
might turn out better, or worse. This risk has been
tried, and found to be in favour of the offices; but in

4

ON THE NATURE OF INSURANCE. 265

another speculation, of another kind, the same species
of risk might give a contrary result.

Among the sources from which the insurance offices
have drawn profit, we must reckon lapsed policies. It
has frequently happened that an individual insuring his
life has continued to pay the premiums for a few years,
and then, either through incapacity to continue the
payment, or because the object of his insurance was
otherwise attained, has allowed his policy to lapse to
the office by non-payment. The office, of course, is
benefited, but not, as might be supposed, by the total
amount of his premiums. What they have received
does not all become profit by the lapse of the policy,
but only that portion by which the premium for the
whole life exceeds the premium for a temporary in-
surance. Every premium which is paid by an insurer
contains the consideration given for the chance of his
dying in each and every subsequent year. If, then, he
remain a member of the office, and stand the risk of
death during a certain number of years, all such part of
his premiums as was consideration for the risks of those
years became due to the office, and was taken by the
office, as compensation for those risks, and cannot
therefore be said to fall to them as profit upon the lapse
of the policy. Two individuals, A and B, go to the
office on the same day, and insure their lives for the
same sum, A upon his whole life, and B for seven years.
“A pays, say 10/. of premium, and B 7/. At the end of
seven years, A allows his policy to lapse, just at the
time when B’s policy expires by its own construction.
What does the office gain by the lapse? Evidently the
temporary annuity of 3/., by which the two: premiums
differ. The 7/. paid by A out of 10/. is not more than
sufficient to pay his share of the claims which arose
during the years which he continued in the office: the
remaining 3/. was a reserve for future years, which
becomes profit to the office on his declining to stand
the risks of those years,

Perhaps no part of the subject is less understood than

266 ESSAY ON PROBABILITIES.

this. Persons having insured for their whole lives, and
being afterwards desirous to discontinue, are surprised
to find that they cannot get for their policies even as
much as the amount of their premiums, to say nothing
of interest. Each of them reasons thus : — Since I did
not die, the office lost nothing by me, and, as it has
turned out, ran no risk: why, then, should they not
restore me the premiums which I have paid? ‘To which
it should be answered: Because the risk, which turned
out favourably in your case, did not produce the same
result in another case; and it is the very essence of an
insurance office, that those who live pay for those who
die. If you can induce the executors of those who have
died during your tenure of your policy to refund what
they have received from the office, with compound
interest, when the office will repay you your premiums,
also, with compound interest. The above-mentioned
reasoning of the insured party is much on a par with
that of the judge in Godsal’s case.

A respectable weekly newspaper has lately allowed
the following doctrine to be promulgated in its columns ;
namely, that it is an undeniable fact, demonstrable by
the books of an insurance office, that very much the
larger portion of their profits has always arisen from
lapsed policies! Till I saw that article, I could hardly
have believed that even 'a newspaper would have admitted
so palpable a mistake. On the supposition (no matter
how false) that all the back premiums of a lapsed policy
are, as they say in book-keeping, to be carried to profit
and loss, how could such an assertion be made, in the
face of the well-known fact, that premiums are deduced
from a table of much higher mortality than that actually
experienced? Those persons, who, one with another,
were expected to live twenty years, have lived twenty-
four years. A small proportion of them have allowed
their policies to lapse, enough to give, perhaps, a per-
ceptible profit to the office, but not enough materially
to increase its funds ; for it must be remembered that,
though the number of policies allowed to lapse bears a

MANAGEMENT OF AN INSURANCE OFFICE. 267

proportion to the whole which might give some colour
to the preceding assertion, yet the value of these policies
is generally small. It is seldom that a policy is aban-
doned which involves a large sum, or on which many
premiums have been paid. If, instead of comparing
the number abandoned with the whole number of poli-
cies, we were to calculate the value of those policies,
and compare them with the value of all the liabilities
of the office, the former would be found a very small
portion of the latter. It is well that it has been so, for
this source of profit is diminishing as the subject.
becomes better understood. It is known that a policy
of a very few years’ standing is worth something, and had
better be sold at any price than abandoned.

All that precedes has reference to the relation in which
the office stands to the public, and to the collective body
of the insured. All dangers, and all remedies, have been
considered merely with reference to the general security
of the establishment, and without inquiring into the
effect produced on the relative interests of the insured.
Since it is the first principle that no interest of one or
the other class of insurers must be consulted to the de-
triment of the whole, the order of .discussion which I
have followed is necessary to the subject. It now re-
mains to treat of the internal management of an office,
and to this subject I proceed in the next chapter.

CHAPTER XII.

ON THE ADJUSTMENT OF THE INTERESTS OF THE
DIFFERENT MEMBERS IN AN INSURANCE OFFICE.

Tuerz is not a circumstance against which it is necessary
to guard in the general management of an office, but what

268 ESSAY ON PROBABILITIES.

is accompanied by this inconvenience, that the measures
adopted, whether of precaution or remedy, may be made
to press unequally upon the different classes of insurers.
If we take, for instance, a fixed rate of interest, suffi-
ciently below that which can really be obtained, we find
that many of those insured must pay their premiums at
a time when interest is comparatively higher, and vice
versd. With regard to the tables of mortality, most
probably (it has always so happened) a table which is
generally too high will be unequally too high; so that
some classes of insurers will contribute more largely to
the safety fund than others. And even in the distri-
bution of the profits, however good the will may be to
apportion them duly, there are yet practical difficulties in
selecting an equitable method out of those which do
not require calculations of insupportable minuteness.

It will only here be necessary to dwell upon two
points, the distribution of the premiums, and the method
of appropriating the profits.

In the last chapter, in speaking of the use of too
high a table of mortality, as a safeguard, I was merely
considering the collective security of the office. There
are two different ways of answering the same end:
either by using a table of mortality confessedly too high,
or constructing premiums from a true table of mortality, —
and increasing these by such a percentage as will pro= —
duce the same receipts to the office. For general security,
these two plans are equally good ; but they may produce ~
very different consequences upon the relative state of the
members. For instance, the Northampton table, which
is the basis of most of those now in use, is certainly, as
already noticed, too favourable to the older lives. Mr. |
Morgan gives the following table*, exhibiting the num- ~
ber who did die, and those who should have died, if the
Northampton table had been correct, all in the twelve
years preceding 1828.

* View of the Rise and Progress of the Equitable Society, London, 1828,
page 42.

MANAGEMENT OF AN INSURANCE OFFICE. 269

Age. Number. Of whom | Of whom should
diddie. | have died.
20 — 30 4,720 29 68
30 — 40 15,951 106 243
40 — 50 27,072 201 506
50 — 60 23,307 339 545
60 — 70 14,705 426 502
70 — 80 5,056 289 290
80 — 95 | 701 99 94

From this comparison, Mr. Morgan concluded that
the superior vitality of the young and middle ages was
the effect of selection, which wore out, so to speak, after
the age at which no new members were admitted ; thereby
proving, in his opinion, at once the effect of selection,
and the excellence of the Northampton table. Now, it
obviously cannot prove both of these things: granting
the latter, it would certainly go a great way to prove the
former ; and granting the former, it does not impugn
the latter: which is all that can be said. But, if it
should happen that the mortality of the Northampton
table is near the truth at the older ages, and very much
above it at the younger, the sort of result shown in the
preceding comparison would follow of course ; and this
circumstance, demonstrated as it is by other and inde-
pendent tables, is, no doubt, the true explanation.

If such be the case, where is the fairness of using a
table which demands premiums very much larger than
the real risks from the young, while it admits older lives
on more easy terms? Ought the older lives to enjoy
any privilege in this respect P Quite the reverse ; for,
(page 253.) belonging to a class which is less known,
and entering also in smaller numbers, with results there-
fore more subject to fluctuation, the percentage, added to
the premiums deduced from a true table, ought rather to
be larger in the case of old lives than'in that of young
ones. The best customers, both in number and quality,
ought not to come worst off.

270 ESSAY ON PROBABILITIES.

The proposed table of Mr. Finlaison (page 259) affords
a striking illustration of this point. It is accompanied
by a table representing the average premiums of all the
offices. At the age of thirty, Mr. Finlaison proposes
to demand 17 per cent. less than the average of what is
now asked by the offices ; at the age of 60, this same
able and strenuous advocate of reduction would only
reduce the average premium of the offices by 34 per
cent. I now put down the present value of 100/., pay-
able at the end of the year in which a life drops, from
the Northampton and Carlisle tables, at 3 per cent.,
and for different ages, together with the percentage
which must be taken from the former to reduce it to the
latter. |

Age. | Northampton.| Carlisle. | Percentage of
difference.
20 £ 42°8 £ 33°9 20°8
30 47°8 40°0 16°1
40 53°8 47°1 12°5
45 57°2 50°8 11°2
50 60°9 55°4 9°0
55 64°6 60°9 B°7
60 68°6 66°5 ra |
65 72:9 411 2°5

In offices, then, which continue to use the Northamp-
ton table throughout, the safety rate is levied upon those
who enter at the age of 20, to the amount of 21 per cent.
out of the total sum they pay; while on those aged 65
it only amounts to 24 per cent. The Carlisle table
represents the experience of the Equitable Society very
nearly.

Again, the Amicable Society now charges premiums
deduced from its own experience, and in which the
fundamental inequality of the Northampton table is
corrected. It will be worth while to compare the ayer-
age of all the offices given by Mr. Finlaison, with the

MANAGEMENT OF AN INSURANCE OFFICE. 271

present premiums charged by the Amicable. The sup-
position is for 100/. insured.

Age. Average. Amicable. Mr. F.'s pro-
‘posed Premiums.

20 £2°02 £2°03 #£1°76

30 2°50 2°53 2°07

40 3°26 $25 2°78

50 4°47 4°83 4°06

55 5°38 5°90 5°00

60 6°58 7°33 6°25

From such comparisons as the preceding, I have long
been of opinion that, safe as the offices are, each con-
sidered as a whole, the proportions of the premiums
demanded at different ages are, in the first instance,
inequitable. To a certain extent, the young are made
to work for the old; that is to say, the person who
insures early in life, the more prudent of the two, is
made to pay a part of the premium of the one who
does not begin till he is old.

The evil is not so great as it might at first sight
appear, for two reasons: firstly, because those who enter
at the older ages are few in number compared with
those who begin between 30 and 50 years of age;
secondly, because many offices make compensation to the
younger members in the division of the profits. Still,
however, the inequality is of a sufficient magnitude to
demand alteration, which will be brought about in an
obvious way; namely, by the younger insurers giving
the preference to those offices in which, premiums and
returns considered together, the inequality is the least.

There is another point, though not of so much con-
Sequence, in which an inequality falls more heavily
upon the young than upon the old ; namely, the method
of paying the expenses of management. The yearly
contribution of every member to this fund ought to be
the same. Suppose, then, that from every premium a
given sum is subtracted, to answer this end, the in-

272 ESSAY ON PROBABILITIES.

equality of the remainders is increased ; it being obvious
that any disproportion which exists between two numbers
is made larger by taking away the same from both.

The way to correct the inequality, without altering
the actual receipts of the office, is as follows. The pro-
portions in which the different ages exist in the office at
any one time can be pretty nearly found. Let the office
table of premiums be taken, and from it let an average
premium be formed, by taking into account as well the
several premiums, as the numbers who pay them. Suppose,
for instance, that A persons pay the premium a, B pay
b, &c. &c.; then the average premium is found by
dividing the sum of the products of A and a, B and 3,
&c., by the sum of A, B, &c. Let the actual average
premium be called P; and let the average premium,
formed in the same manner from a true table of mor-
tality (in which a, b, &c. are different, but A, B, &c.
the same as before), be Q. Let P exceed Q by & per
cent. of Q ; then the premiums given by the true table,
increased by & per cent-, are those which should be sab-
stituted for the existing premiums, in order that all
inequalities may be corrected, without diminishing the
receipts of the office. It matters nothing, in the pre-
ceding rule, whether the premiums of what has been
called the true table are correct or not, so long as their
proportions are correct; and one office might, by this
rule, adopt the proportions of another, without altering
its own receipts.

If such a process as the preceding were performed,
deducting from the receipts required by the office the
whole expense of management, and afterwards adding
the last-mentioned item in equal shares to all the poli-
cies, the distribution of the premiums would be theo-
retically perfect. It remains to consider the more difficult
part of the question,—the method of dividing the profits,

Hitherto, I have had no occasion to speak of a most

important difference of system which distinguishes one

office from another ; the distinction of mutual and pro-

prietary. The former have no capital, except what arises

MANAGEMENT OF AN INSURANCE OFFICE. 273

from their own accumulations, and each member is a
guarantee to the rest for the fulfilment of all engagements.
If the office possess a charter, this guarantee operates no
further than to pledge the premiums already paid by any
member for the discharge of all claims which arise be-
fore his own, since a corporation is considered in law
as an individual. If, on the other hand, there be no
charter, the whole fortune of every member is pledged for
the discharge of all claims. The risk, however, at the com-
mencement is not great in character, and small in amount ;
and the quantity of risk diminishes so much faster than
the amount increases, that it may safely be said there is
nothing in the commercial world which approaches, even
remotely, to the security of a well established and pru-
dently managed insurance office.

A proprietary insurance office has a capital, the
proprietors of which may or may not be insured in
the office, and for which such a bonus is paid, in ad-
dition to the market rate of interest, as is mentioned
in p. 263. It would perhaps be difficult, at the pre-
sent time, to establish a new proprietary office with a
very large capital. The public now begins to see
that much capital is not necessary, and that nearly
all the bonus which is paid for its use is so much taken
away from the savings of the insured, without any ade-
quate benefit received in return. One by one, the pro-
prietary offices must (as some have done) admit the
insured to a share in the profits: the necessity for which
will be taught by the decline of business, if not previ-
ously learnt.

The question as to how profits should be divided, is
of the same nature in both species of offices; the differ-
ence being, that the offices which are partly proprietary
have less to distribute among the insured than those
which are mutual. The first inquiry must be, What is
the profit of an insurance office; and how is the amount
to be ascertained? Firstly, as to the profit which an
insurance office may be expected to realise, judging by
the premiums they receive, and the mortality they have

T

274 ESSAY ON PROBABILITIES.

hitherto experienced. Certain limits may be obtained
which may sometimes serve as a useful check.

Perhaps the average age of admission to an insurance
office is about 40, as many entering younger as older.
The average premium charged by the offices at that age
is 3-26/. per cent. Now the most extreme supposition
which can be imagined in favour of the insured is, that
the Carlisle table should be taken as the law of mortality,
and 4 per cent. as the interest of money. Upon these
suppositions, the accumulations of the office would
amount, upon a premium of 3°26/., at the death of
parties aged 40, one with another, to 137/. But this
pushes every favourable supposition to its extreme, and
moreover allows nothing for expenses of management.
I am inclined to think, however, that the usual pre-
miums will, as long as the rate of mortality continues at
its present amount, yield about 125/. for 100/. nominally
insured, and perhaps something more. |

It must not be left out of sight, that the offices con-
sider every person as having the age which he will attain
at his newt birth-day. If, for instance, a person who
attains 40 years of age on the 12th of March were to
insure his life on the 13th, he would be said to be 41
years of age, and would have to pay accordingly. The
effect of this very proper* regulation is, that, one party
with another, all are half a year younger than their office
age. Again, all the tables are computed on the supposition
that interest is made yearly, whereas in fact it is made
quarterly. Circumstances of this sort, trivial as they
appear, do nevertheless produce a sensible effect in a
jarge number of years. To the above we must add the
profits arising from the purchase of policies, which is
always done by the offices on terms very favourable to
themselves ; fines for non-payment of premiums; the
profits of lapsed policies ; and so on.

Leaving all speculation as to the probable profits,
I now proceed to show how to ascertain, from the

* Proper as long as there is no subdivision of a year. I think the offices
might very rationally divide the year into quarters.

MANAGEMENT OF AN INSURANCE OFFICE. 275

actual statistics of an office, what its real condition
is. And here I must observe, that though in the con-
struction of premiums, a table of more than the real
mortality must be used, yet no such thing is absolutely
necessary in the valuation of its liabilities and assets.
Here truth, and not security, is the object ; and if by
any means a true table can be obtained, its results
should be calculated ; though I do not say that in the
declaration of profit, such results should be admitted
to their full extent. The most simple theoretical way
of conducting the process, is to ascertain the value of
every policy, as in page 218. ; that is, to ascertain how
much should be given to the holder of each policy to
renounce his claim, the office also abandoning the future
premiums. When this is done, it is obvious that the
office is not solvent, unless the assets arising from the
accumulations of former years be sufficient to pay the
values of all the policies, and thus to buy them all up.
Supposing the office able to do this, with a capital
remaining larger than would be necessary to create a
permanent fund for the expenses of management, the
surplus of that capital is profit. Otherwise, calculate
the present value of all premiums due to the office, and
also the present value of all claims to which it is liable.
To the former add the sum total of the assets of the
office, and to the latter add the present value of a per-
petuity equal to the expenses of management. Thus, let

P = present value of all premiums.

C = present value of all claims.

A = total assets of the office.

M = present value of all expenses of management.
If then P and A together exceed C and M together,
the office is solvent, and the excess is profit.

On each of these items a few remarks may be made.

(P.) All the parties who are of the same office age,
may have their several policies considered as one col-
lective policy, in respect of which the sum of the
premiums is paid as one premium, and the sum of the

7%

rw

76 ESSAY ON PROBABILITIES.

possible claims is one claim. But as these premiums
are payable at all periods of the year, they may be con.
sidered as, one with another, due at six months after
the valuation, at which time the present office age of
the parties may be considered to be their real age.

(C.) All bonuses which have actually been added to
policies (if any) must be included in the claims; and
the value of each claim must be carefully found, with
reference to the time after death at which it is paid.
(See Appendix the Second.)

(A.) The principal of the assets must be deduced |

entirely by means of the income it yields, and must be
ascertained from the income by means of the rate of

interest assumed. On this subject, which contains a dif. —

ficulty of a peculiar character, see the Sixth Appendix.

(M.) Against the expenses of management may be
set, as far as they go, the incidental profits, when they —

can be tolerably well ascertained.

The profit being thus found, and that share of it
which belongs to the insured (if the office be not

mutual), it remains to inquire, What principle of division
should be adopted? And, firstly, it may be doubted —

whether the whole of the profit i is immediately divisible,
consistently with prudence. To use an astronomical

phrase, the increase of the surplus is partly secular, and
partly periodic; that is to say, instead of a steady and

uniform increase, there is a fluctuating rate of aug-—
mentation, compounded of that permanent rate which
the largeness of the premiums necessarily gives, and
the alternate accelerations and retardations occasioned —
by the departures of the incidents of the several years
from the average. The only way of obtaining the per-
manent part of the surplus is by estimating it on the —
average of a considerable number of past years, regard

Sodas’ Sah Saale ———— Se “Pe

being had to the relative, not the absolute, surplus. —
Let us suppose, for instance, that the present value of o

all claims is ascertained to be one million, and the present —

value of all premiums 700,000/., the office possessing

:

besides (clear of charges of management) 500,0001.: —

iv
¥
a
¥
4

MANAGEMENT OF AN INSURANCE OFFICE. 277

_ there is then a surplus of 200,000/. ; which having been
- accumulated out of premiums, and profits having been
regularly paid up to the present time, it may be pre.
sumed that the premiums themselves are capable of
maintaining this rate of surplus. The office must then be
presumed able to pay 120/. for every 100/. insured.

But it is important to note, that the present rate of
profit must not always be assumed as that which can be
permanently maintained. Suppose, for instance, an office
which begins for the first time to divide profits: its
accumulations are therefore, in part, the reserves of profit
which should have been added to former claims, had any
division of surplus previously existed. The same remark
may be necessary when any change is made in the way
of dividing profits, since the surplus existing at the
moment of the change is the result of a former state of
things. Thus, an office which has proceeded injudici-
ously, in making too large divisions, may possibly, when
it adopts a more prudent system, be justified in forming
a system which would require a larger surplus than the
one which it actually possesses at the time of discover-
ing the error; for the then existing surplus has been
unduly weakened, and is not to be considered as repre-
senting the permanent effect of the improved mode of
proceeding. |

I have stated, that the percentage which can be added
to each 100/. insured should be determined by the
average of a number of years. If this number be too
great, the incidental fluctuations of mortality may be
compensated ; but at the same time the real and secular
changes of mortality may be prevented from producing
their proper effect. As long as the value of. life is in-
creasing, too long an average is a defect on the safe side:
but if it were diminishing, it might happen that the
mean of a number of preceding years would present a
higher result than would be consistent with security.
As yet the offices have had nothing to encounter except
the diminution of mortality, and its consequences ; but

t 3

278 ESSAY ON PROBABILITIES.

in constructing rules for their own guidance, they should
be careful not to fall into such errors on the safe side
as become errors on the wrong side when circum-
stances change. I hold an opinion which I think, from
his writings, was also that of the late Mr. Morgan;
namely, that an insurance office must consider the last
half century as having been a period of circumstances
singularly favourable for the formation and growth of
such institutions, more so than it would be wise to expect
for the future. Perhaps from five to ten years is the
length of time for which the preceding average should
be computed.

The valuations should, if possible, be made yearly.
No check which can be devised is so likely to be useful
as yearly valuation ; and it is absolutely necessary to any
system which gives the real amount of their premiums
to the insured. In a mutual insurance office, starting
without much capital, it would be madness to rest upon
any tables and to neglect valuations ; unless, as before
remarked, the returns made to the insured are meant to
be very much below their payments. And in conjunction
with yearly valuations should come yearly divisions of
profits, or something equivalent. There is, 1 believe, a —
prejudice against frequent divisions in the minds of |
many who have derived their ideas on the subject from —
the former practice of offices. But surely, provided that
the proper amount of profit be divided yearly, and no ©
more, it matters nothing whether the apportionment be —
made seven times in seven years, or once only, as far as
security is concerned. For it is to be remembered, that
yearly division of profits does not imply an annual expen~-
diture, but only an annual distribution of future expen-
diture. In septennial divisions, one of two things always
takes place: either the profits are made contingent upon
a party surviving one or more periods of division, which
creates great inequalities between the lot of different
persons (the very thing an insurance office was intended
to avoid); or it declares beforehand, what the profits
shall be during periods of seven years. In the latter

MANAGEMENT OF AN INSURANCE OFFICE. 279

case the annual division is unquestionably the more safe ;
since it is easier to predict the capabilities of one year
than of seven.

In writing upon any point connected with insurance,
the practice of the Equitable Society naturally suggests
itself. Nevertheless, I always consider that society as a
distinct and anomalous establishment, existing at this
moment under circumstances of an unique character.
It is the result of an experiment which it was most
important to try ; but which having been tried, need not
be repeated. Its history is briefly this: The Ami-
cable Society, which, in the year 1760, was the only
one existing, was originally founded rather on princi-
ples of mutual benevolence, than of mutual insurance,
as now understood. A certain number of persons (the
only restriction being that their ages should be between
twelve and forty-five), each paying the same sum
yearly, the whole fund of each year (or the greater
part) was divided among the representatives of those
who died within the year.* The Equitable Society
was founded upon the principle of apportioning the
payments to the risk of life. The stables were con-
structed by Dodson, who, as Mr. Morgan remarks,
“¢ for greater security assumed the probabilities of life
in London, during a period of twenty years ; which,
including the year 1740, when the mortality was almost
equal to that of a plague, rendered such premiums
much higher than they ought to have been, even ac-

cording to the ordinary probabilities of life in London
itself.” The truth of this remark will sufficiently ap-
pear, from comparing the average of the present office
premiums with the original Equitable premiums, as
given in the following table. And even these pre-
miums were increased on the most frivolous pretexts.
Thus female life and young life were considered as
more than usually hazardous, and paid for accordingly.

* The Amicable Socicty now retains only one of its original characters ;
namely, that all members, whatever may be their age at death, or the term
of their continuance in the society, participate equally in the profits.

Tt 4

280 ESSAY ON PROBABILITIESe

Age Equitable Equitable | Average present
* |Premium, 1771.)Premium, 1779.| Premium.
£: -a.; 0: Lies & ££: ad
14 9.450 IE, RS =v as
20 a. 9 * 212 10) 2 0.0
25 314 0 o aS. RN FEMS
30 338° 7 > rns 2 10
40 417 9 4 711 Ss oO
49 6:2 :§ 6 10:3 4.6: Q

Mr. Morgan says, “that for the first twenty years,
the society possessed such an excess of income, that
being suffered to accumulate without interruption, it
contributed, in a great measure, to form the basis of its
future opulence.” This circumstance, with the great
number of policies which were abandoned* in the
early stages of its career, and the increase of interest
during the war, are quite sufficient to explain the
wealth which the Equitable Society has accumulated :
to these must be added the parsimony with which, at
first, additions were made to the policies. The whole
was an experiment, on a graduated scale of premiums,
made with a caution, which, though it turned out to be
superfluous, could not be known to be such, except by
the result. It was at the same time a venture, and by
many considered as a hazardous one ; for instance, the
law officers of the Crown refused acharter, on account of
the lowness of the premiums. The hazard having been
run, and having turned out profitably, the proceeds be-
long to those who ran it, and to those who, by their
own free consent, became their lineal successors. Nor
is it the least remarkable circumstance connected with
this society, that the immense funds at its disposal have
been always opened, though under restrictions, to the
public. Though this has been done in a way which
renders the participation of the new insurer in the

* Perhaps Mr. Morgan’s statement on this point may have led to the
statement alluded to in page 266.

MANAGEMENT OF AN INSURANCE OFFICE. 281

previous accumulations a remote contingency, still it
is done, and by a body who might without any bar,
legal or moral, immediately close their doors, and divide
the whole among themselves.

I have made the preceding remarks, in order that it
may be clear how little the history or practice of the
Equitable Society should have any direct authoritative
bearing on the spirit in which the management of a more
modern office should be carried on. The general lesson
taught by it is, — be cautious ; but, among other things,
be cautious of carrying caution so far as to leave a part
of your own property for the benefit of those who are
in no way related to you. If there be a Charybdis in
an insurance office, there is also a Scylla: the mutual
insurer, who is too much afraid of dispensing the profits
to those who die before him, will have to leave his own
share for those who die after him. Reversing the
fable of Spenser, we should write upon the door of
every mutual office but one, be wary ; but upon that one
should be written, be not too wary, and over it, “‘ Equit-
able Society.”

An insurance office has no existence separate from
that of its insurers; and no public duty to fulfil, ex-
cept to collect, improve, and equalize their premiums
(p. 238.): therefore, their most important object,
next to the fulfilment of their guaranteed engagements,
is the distribution of their profits in such manner that
every one may obtain his due share. The question now
becomes, What is the due share of each party ? This is,
in some measure, a question of previous contract, though
there are those who consider that there must be a right
and a wrong way. For instance, Mr. M‘Kean, the
compiler of the tables alluded to in page 191., and of a
useful work * which accompanies them, says, ‘ Our
conclusion, and a most important one, lies conspicuous
on the very surface. It is impossible that atu the

* “ Exposition of the practical Life Tables, &c. London: Butterworth,
Richardson, &c. 1837.’? This workis, I believe, sold separately.

282 ESSAY ON PROBABILITIES.

offices above mentioned can be correct or just in their
aws for dividing the surplus. If the plan of the
Equitable is right, then most unquestionably the plan of
the Atlas is wrong, and great injustice is done to the
younger members, and so vice versd. But, is this a state
of things in which so important a system as that of life
insurance, based, as that system is, on mathematical
science, ought or can continue to exist ? Certainly not.”

On this I observe, that though life insurance be an
application of (not based upon) mathematical science,
yet that the entrance of exact numerical reasoning is
subsequent to the admission of certain principles, and the
experimental acquisition of certain facts. It is not by
mathematics we learn that life is uncertain in individual
cases, but nearly certain in the mass — that it is the
duty of every one to provide for his family — and that
this can be done without contingency, if those who
survive the average term agree to surrender a part of
their substance to those who do not. Calculation will
point out the amount which, upon any given principle
of division, belongs to one or another of the insured ;
but before we can come to this point, it must be settled
with what intention the surplus was paid ; which may
be different in different offices. The following con-
siderations might be addressed to any person who in-
tends to insure his life: — You are aware that the pre-
mium demanded of you is, avowedly, more than has
hitherto been found sufficient for the purpose, the reason
being, that it is impossible to settle the exact amount, on
account of our not knowing whether the future and the
past will coincide in giving the same law of mortality,
and the same interest of money. ‘The surplus arising
from this overcharge, for the future existence of which
it is hundreds to one, is now at your own disposal, and
you must choose between one office and another, accord-
ing to your intentions with regard to its ultimate des-
tination. Firstly, if you doubt the general security of
the plan of insurance, and are desirous of an absolute
guarantee, independently of accumulations from pre-

MANAGEMENT OF AN INSURANCE OFFICE. 283

miums, there are offices which will, in consideration of
the surplus aforesaid, pledge their proprietary capitals
for the satisfaction of your ultimate demand upon them.
Secondly, if, being of the opinion aforesaid, you think
the whole surplus too much to pay for the guarantee,
there are proprietary offices which retain a part of the
profit in consideration of the risk of their capital, and
return the remainder. Thirdly, if you wish the surplus
premium, as fast as it is proved to be such, to be ap-
plied in obviating the necessity of any further over-
charges, there are offices which divide the profits during
the life of the insured, by means of a reduction of pre-
mium. Fourthly, if you wish the surplus to accumulate,
and, feeling confidence in your own life, are willing to
risk losing it (the surplus, remember) entirely if you
die young, on condition of having it proportionally in-
creased if you live to be old, there are offices which
divide all or most of the profits among old members.
Fifthly, if you would prefer a certainty of profit, die
when you may, there are offices which at once admit
new members who die early to a full participation in all
advantages. The choice between these several modes
must be made by yourself, according to your own
inclinations, views of fairness, or particular circum-
stances,

There are three modes of division which deserve par-
ticular notice; namely, periodical additions to the policies,
periodical diminutions of premium, and addition to the
policy at death to an amount depending upon the assets
of the office, without reference to the time during which
the insured has paid premiums. I may, perhaps, be
thought to treat this subject with prolixity ; notwith-
standing, knowing that this part of the subject has
created more discussion of late years than any other, I
think an attempt to compare the principles of different
plans not out of place.

The considerations- which follow will apply to all
offices which divide any profits whatever: the inquiry
being, not how much surplus should be divided, but in

284 ESSAY ON PROBABILITIES.

what proportions a given sum should be divided among
the insured.

Let us return to the original constitution of an in-
surance office (page 238.), derived from the statement
of its main object; namely, that it is a savings’ bank
with a power of equalizing those results in which the
different durations of life would cause differences. Sup-
pose that such an office sets out with premiums imagined
to be no more than sufficient, but which are afterwards
- found to be more than sufficient, leaving an admitted
amount of surplus in hand. The first thought would
be of restitution; namely, rendering back to each indi-
vidual the amount which he had bond fide contributed
towards the surplus. To do this properly, it must first
be settled whether the insurance office is one or many.
Does each age insure itself, or do the separate ages in-
sure both themselves and each other? If the premiums
were properly proportioned, there would be no occasion
to ask this question: but if the incomers of one age pay
unduly as compared with those of another, then it is but
fair that they should receive in proportion. In the dis-
tribution of premiums, which I have described in p. 270.,
it is equitable that a remedy should be provided, by
virtue of which those who enter the office young should
receive more than the rest. And it is, for this reason,
desirable that the proportions of the division should be
regulated by a true table of mortality.

Let P be the real premium, and P + p the office pre-
mium ; and let the death of an individual take place
after he has been m years insured, and just before the
(n+1)th premium is paid. If the office had been a
compound interest savings’ bank, the deceased would,
at his death, have been entitled to the following amount.

P + p improved at compound interest for n years
OO a ae ee ae ee

BAN 4 0) a ike oh Chane ree wee ee
But under the conditions of insurance, the part P, with

MANAGEMENT OF AN INSURANCE OFFICE. 285

its accumulations, is the consideration for the sum in-
sured ; the remaining part p, with its accumulations, is
due under the name of profit or restitution, in a strictly
mutual office.

The application of the preceding method would require
that a calculation should be made once in every year of
the quantity p and its accumulations, for every individual
insured. This having been done, and the surplus
A+P—C having been calculated from a true table of
mortality, it is then known in what proportion any two
individuals insured are claimants upon this fund. Sup.
pose that p and its accumulations amount, in the case of
the persons X and Y, to 100/. and 1507. Suppose that
A+P—C is 100,000/., and that the sum of all the
excesses of premium with their accumulations, of which
the 100/. and,150/. just mentioned are items, is 120,000/.
It matters nothing that the last sum is greater than
100,000/., since we are not speaking of a fund on which
there are definite claims, but of one the nature of
which it is to be of uncertain amount. The use of the
items 100/. and 1502., and of the sum total of 120,000/.,
is to enable us to divide the real fund of 100,000/. among
those who raised it, in the proportions in which they
contributed towards it. Thus if X and Y were to die
in the year of the valuation, it would be fair that they
should receive such proportions of the 100,000/. as 1007.
and 150/. are of 120,000/.; that is, five-sixths of 1004.
and 150/. This method proceeds upon the principle that
all the excess of premium is taken in trust as a guarantee
for the main fund, and is to be returned if not wanted, or
such proportion of it as is not wanted. It confines the
insurance, or provision against the uncertainty of life,
entirely to a stipulated sum, and regards all that part of
the premium which is not really wanted to provide this
sum, for one man with another, as paid into a com-
mon savings’ bank, in which no equalization is supposed.

The labour of making the calculations would, I ima-
gine, prevent any office from adopting the preceding

plan, so as to carry it into execution yearly. Witha

¢

286 ESSAY ON PROBABILITIES.

good system, however, the difficulty of managing the
details of such a scheme would not be so great as at
first sight might be supposed. Upon its principle hang
the two first plans of division mentioned; namely,
periodical additions to the policies, and periodical di-
minutions of premuim. In both of these, the advantage
of the insured is increased by the length of his life ;
that is to say, the excesses of his premiums are placed ©
to his credit in the first, and considered as having been
prospective payments of his future premiums in the
second. But nevertheless there runs through the offices
which adopt these plans more or less of a practice which
prevents the surplus from being divided among the in-
sured in equitable proportions. Suppose that there is a
septennial bonus, as it is called, which was declared in
the year 1830. Immediately after the award, two
persons, A and B, aged 30 and 60, enter the office each
upon a policy of 100/., and were both alive when the
bonus of 1837 was declared. This bonus is generally a
percentage, not upon the amount of premiums paid, but
upon the sum insured, and both would have the same
addition made to the 100/. for which they have insured.
But have both contributed to the accumulations of the
office in the proportion which would render this mode of
division equitable? To consider this point, remember
that a promise to pay, say 5/., at the death of a person
aged 67, is of much more value than the same at the
death of a person aged 37. The older life therefore
receives much more than the younger life. But he has
paid much more. That is true ; but at the same time
he has occasioned a greater risk to the office, and it is
the excess of his premium above the risk (and not the
whole premium) which the office acknowledges in de-
claring the bonus. From page 270. it sufficiently ap-
pears that the premiums of the older ages are already
too small in comparison with those of the younger:
this mode of dividing the surplus, therefore, only tends
to increase the existing injustice. The only remedy is,
to make use of the process laid down in the preceding

MANAGEMENT OF AN INSURANCE OFFICE. 287

page ; and having ascertained the amount of what each
person has paid over and above what was necessary, to
consider each person as entitled to the sum which his
overplus would purchase at his death, if the bonus be
made by addition to his policy ; or to a diminution of
premium answering to the annuity on his life, which the
overplus would buy, if the bonus be made by diminu-
tion of premium.

The knowledge, therefore, of the real premium is
necessary for an equitable distribution of the surplus,
upon the supposition that the said distribution is made
on the principle of dividing the surplus fund among the
contributors in proportion to their contributions. Every
plan which, ceteris paribus, makes equal additions to the
policies of different ages, is inequitable. I repeat again,
that in the preceding cases, the principle of division
ought to be considered as arising from the combination
of an insurance office and a savings bank ; the por-
tion of premium which covers the risk of life being
paid to the former, and the remainder to the latter.

The third method of division supposes the establish-
ment to be entirely an insurance office, and not at all
a savings bank. Its object is to make the returns to
the different members both equal and equitable. Con-
sidering that the real risk of life is not perfectly ascer-
tained, and that if it were it would not be safe to re-
duce the premiums to the lowest theoretical safety-point,
such an office, instead of demanding a premium avowedly
too high for the sum insured, and engaging to return
all or part of the surplus, considers the sum insured as
indefinite, except only in so far as a minimum is named,
below which it is not to fall. Thus such an office,
receiving, say 31. of premium, from a person aged 34,
for what is called, in compliance with custom, a policy
of 100/., does in fact make the following bargain: —The
office engages to return, at the death of the party, let
that take place when it may, such a sum as will represent
the average accumulation of an annuity of 3/. continued
during the life of a person aged 34, be that sum more

288 ESSAY ON PROBABILITIES.

or less ; with this additional limitation, that the office
undertakes that the said accumulation shall not be less
than 100/. This last guarantee, though necessary for
the satisfaction of the public, is in truth so certain, from
the amount of the premium demanded, that a person
acquainted with the subject looks upon the possibility
of the funds of the society suffering from it as an ex
tremely remote chance.

In order, however, to make the proceedings of such an
office equitable, the proportions of the premiums paid by
parties of different ages must be fairly regulated. On
the supposition that the inequality pointed out in page
270. is allowed to exist, the preceding methods of divi-
sion may be (I do not say are) adjusted so that every
interest shall be consulted. But in the present plan, it
is impracticable to remedy any such defect of proportion,
at least without dividing the establishment into as many
different offices as there are ages, which would not be
easy, and perhaps not very safe. The simple rule for
determining the relative premiums is to make them
proportional to the real premiums, with the exception
of a given addition to each (not premium, but) policy,
for expenses of management. In a large office, how-
ever, the expenses of management may be made a part
of the percentage addition to the premiums.

The method of division in such an office is extremely
simple, and has been already described in page 276.
Subtracting the present value of all the claims, that is,
of all the minimum claims, reckoned as 100/. for each
tabular premium paid, from the sum of the present
values of all premiums, and of the assets of the office,
the proportion which this remainder is of the present
value of all the claims expresses the fraction of 1001.
which may be added to each 100/. insured.

Let A, the assets * of the office, be 500,000/.; P, the
present value of all premiums, 600,000/.; and C, the
present value of all claims, 850.000/.: then A+P—C,

* Diminished for the expenses of management, as in page 275.

MANAGEMENT OF AN INSURANCE OFFICE. 289

the surplus, is 250,000/., which being 25 parts out of
85 of the whole claims, or 29=5, per cent., will afford
129,-/. for every 100/., which is guaranteed. Those
who die in the year of this valuation, may therefore
receive that sum.

The principle on which the preceding division is
made, is, that if the same state of things continue,
every one will in turn receive the same dividend. But,
can such a prediction be made? Undoubtedly not,
for the fluctuations, both of those who come into
the office, and those who go out, will tend to produce
variations. It is very unlikely that any office should
maintain itself for a long series of years nearly in the
same position ; and, since the idea of allowing any per-
manent diminution of the surplus must not be ad-
mitted, there is no alternative except an arrangement
for a gradual increase, which it is the object of this mode
of division to make as slow as is consistent with the cer-
tainty of having it. But in this case, it may seem as
if the old system were revived, and a fund instituted
by the present insurers, for the sole benefit of those
who come after them. There is, however, an im-
portant difference between never paying more than the
guaranteed minimum, so that all the surplus goes to-
wards that fund, and drawing upon the surplus nearly
to the full amount which safety would allow, leaving
only such a trifle to augment the fund as is requisite to
avoid too large an out-going. The old principle, then,
which formerly prevented any bonus whatsoever, is here
merely applied to such an extent as to keep the bonus
within proper limits.

If the tables of mortality by which the profits are
divided, be actual representatives of existing mortality,
and if the number of members remain nearly the same,
the indications of these tables, implicitly followed,
would soon reduce the surplus of the office to that
which is barely necessary for the extreme payment
which the premiums will admit. To take a case: sup-
pose that the premiums will in the long run pay 1254/.

U

290 ESSAY ON FROBABILITIES.

for every 1001. guaranteed; the present value of all
the claims is 1,000,000/., that of all the premiums
700,000/., and the value of the assets of the office
600,000/. The surplus is therefore 300,000/., and,
going upon real tables, the office begins to pay 130/. for
every 100/. guaranteed ; and this it would be able to
do in favour of all who are insured at the time of the
preceding valuation, But part of this dividend does
not, and, by hypothesis, cannot, arise from the pre-
miums: it is therefore paid entirely out of surplus,
‘and will gradually disappear. The dividend will be
reduced to 125/., about which it will fluctuate, being
sometimes a little less and sometimes a little more. An
increase of business in such an office would make the
surplus disappear more rapidly, since each new comer
brings in an equivalent to 125/. and those of the new
comers who die receive 1307. A diminution of busi-
ness would produce a contrary effect ; and a total ces._
sation of new comers would allow the dividend to re-
main at 130/. As far as any danger from fluctuations —
of mortality is concerned, I do not see any objection to —
such a division as the preceding: but when it is re-
membered that the possible diminution of the rate of ©
interest must also be provided for, I think it would be —
prudent to reserve a small proportion of the surplus —
for accumulation. 3 |

There are two ways in which this reserve may be —
made ; firstly, by employing a table of less than the —
real mortality in the valuation of the claims and pre-_
miums; secondly, by calculating the surplus from a
real table, and dividing as upon the supposition that a —
given fraction of this surplus, say one eighth or one —
tenth, should be expunged in the calculation. The —
latter plan is the best of the two, in every respect but —
one, as follows. The mutual insurance office must bea —
republic, and many of its members have very little in- _
formation upon the questions which are, from time to
time, submitted to them. They are easily dazzled by
the appearance of surplus, and are quick to believe that —

*

MANAGEMENT OF AN INSURANCE OFFICE. 291

a larger division might be made in their favour. Add

to this that the older members carry with them in the

discussion of questions, that influence which age na-

-turally and properly gives in the management of impor-

tant affairs; and as to which the conduct of an in-
surance office only forms an exception, because questions
arise in which the interests of the old and young
clash * with each other, which is nowhere else the case.
Under such circumstances the disposition to break in
upon the surplus is the fault to which the body has a

q tendency, and it is not a bad thing to place some small

difficulties in the way of doing this. Now if a fraction

of the surplus be withdrawn from the calculation of

the dividend, it is very easy to change one fraction
into another. A vote of the general meeting, and a few

_ strokes of the actuary’s pen, and the thing is done.
- But when the requisite fraction of the surplus is de-

ducted by the supposition of a lower rate of vitality

_ (or of interest) than actually prevails, no change can
_-be made without the entrance of a large number of

important considerations, the discussion of which occu-
pies some time, and places a useful check in the way of
the restless.

But is it then proposed that every office shall be pro-

_ vided with a fund, which, though slowly, is yet inde-

ee ae

finitely, toincrease? Not necessarily; for the reserved

portion of one year is not put aside, and considered as
inalienable, but enters into the surplus of the next year.
There may be, then, a limit to the increase of the sur-
plus, as follows. Suppose the office to be in a stationary

_ state, having arrived at the point where the influx -of

the new members compensates the efflux occasioned by

death or surrender. The receipts of the office consist

entirely in premiums and produce of gapital, the ex-
penditure in management and payment of claims. As
long as the surplus increases, the sum of the first pair
_ * The members of a mutual insurance office are not properly repre-

sented in their list of directors, unless the individuals composing it are of
‘very different ages.

u 2

2G2 ESSAY ON PROBABILITIES.

of items will exceed that of the second; and, whatever —
may be laid by in each year, it produces a larger sur- ~
plus, and larger payments on account of claims, in the 3
next year. If, then, the surplus could increase without
limit, so would the dividends ; but if the surplus have —
a limit, the dividends also have a limit: and it is plain ©
that the limit arrives, when the yearly outgoings from
claims and management are equal to the receipts from ~
premiums and interest of capital, A mathematical in- ~
vestigation of the conditions necessary in order that the ~
fund may increase, but not without limit, gives the ©
following result : :

Suppose an insurance office, constructed upon the ~
preceding principles, to have arrived at its stationary
state, with respect to influx and efflux of members,
and make the following suppositions: 3

A The assets of the office, for precision, say Janu- ~
ary 1, 1838. E

P The veal present value of all premiums from —
members then in existence. 4

C The real present value of all claims (not includ-
ing additions) to which the office is then liable. 4

m The expenses of management till J anuary 1, 1839. —

p The amount which will accrue from premiums and |
su ceeat of premiums by January 1, 1839. 4

e The amount of claims (not including additions
from the surplus fund), which will be paid before
January 1, 1839.

r The interest of one pound for one year. a

t The fraction which is taken of the tabular surplus
fund i in the computation of the dividend.

We suppose (as must be the case in an old office), 3
C greater than P, and (as must be the case in a solvent —
office) A and P together greater than C.

1. In order that there may be a surplus fund in-—
creasing, but not without limit, find the fraction which —
a year’s interest on C is of c. Then ¢, or the fraction
of the surplus fund (or of A+ P—C), which enters
into the formation of the dividend, must exceed that

MANAGEMENT OF AN INSURANCE OFFICE 29%

fraction which a year’s interest on C is of c, otherwise
the fund would increase without limit.

2. Neither can there be such a fund unless the sum
of m and ¢ should fall short of the sum of p, and of a
year’s interest on the excess of C over P. But, when
this is the case, the limiting surplus capital is found
by dividing the excess of the second total just men-
tioned over the first, by a divisor obtained as follows :—
multiply together ¢ and c, divide the product by U,
and subtract r from the quotient. To this surplus
capital, add the excess of C over P, and the limiting
capital is obtained.

8. If it should happen that the limiting surplus
capital is less than the actually existing surplus, it is a
sign that the action of the preceding plan would
diminish the surplus towards that limit instead of
increasing it. In such a case, the surplus is already
too large for the value of ¢ to increase it; and if ¢ be
not diminished, that is, if less of the tabular surplus
be not taken into the computation of the dividend, the
fund will diminish.

It is not to be supposed that any office will ever
reach a stationary state; but the approach may be near
enough to make the preceding process of some use in
the determination of the dividends due to the insured.
If, following the plan which the preceding problem
supposes, we were to inquire what value should be
given to the fraction ¢, the answer to the question must
depend on the reduction of interest which is supposed
within the bounds of probability. Suppose the present
rate of interest to be 34 per cent. and that the extreme
limit is supposed to be 2} per cent, in such a case
the value of P and C must be calculated at 2} per
cent., and such a limiting surplus must be fixed upon
as will, at that rate of interest, enable the office to pay
at least its guaranteed claims. But it is impossible to
lay down an entire system of rules for the regulation of
a species of undertaking which depends on the fluc-
tuations of the state of society. Whatever maxims

u 3

294 ESSAY ON PROBABILITIES.

may be collected, and however sound they may be,

skill and judgment will always be requisite to apply 4

them to the cases which arise. In this respect the
offices resemble the individual problems which arise in
life contingencies. Many as are the cases which have
been described in books upon the subject, almost every
application of them requires attention to some cir-
cumstance peculiar to the instance in question.

CHAPTER XIII.

MISCELLANEOUS SUBJECTS CONNECTED WITH IN-
SURANCE, ETC.

Tue limits of this treatise will only afford a few words
on several points of interest, which I will therefore
condense into one chapter, taking the subjects as they
arise.

The management of annuity offices is somewhat
more easy than that of insurance establishments ; and
the maxims of security in the former are, of course,
the direct reverse of those in the latter, so far as any
considerations of mortality are concerned. ‘Tables
must be assumed of higher than the real vitality, and
a rate of interest somewhat below, or at least not above,
that which can actually be obtained.

Those who wish to buy annuities on the firmest
possible basis, may deal with the government. The
commissioners for the reduction of the national debt
are empowered to grant annuities in lieu of stock, on
terms calculated from the government tables (page 168).
The rates are high; and though a private office may
really be as solvent as the nation, yet confidence springs

Severe Sows Seah Cele ee

" ‘a Ree 83
py ae Rts ae te ES
ego. Sys OR se

MISCELLANEOUS REMARKS. 205

from opinion, and the security of the national debt
must always be thought the very best. The patriotic
annuitant, too, may reflect that the profit derived from
him goes to the reduction of the national debt.

The distinction of male and female life becomes of
importance in the granting of annuities. The insurance
offices have not as yet, except, I believe, in one or two
instances, begun to recognise the distinction, which is
of the less consequence, since, with respect to the
office, it is keeping on the safe side, and, with respect
to the public, very few female lives are insured. But
the exact reverse takes place with regard to annuities ;
it would be insecure to grant them to females on the
same terms as to males; and a very large proportion of
the whole number of annuitants is of the former sex.

Annuities might be granted by an office which
should undertake a return of profits, in the form of a
payment to the executors at the death of the party ;
and an association of mutual annuitants would not be
of difficult formation. The principal objection would
be, the smallness of the number of persons who buy
annuities, compared with those who insure their lives.
If, however, such an office were to grant reversionary
annuities, their field would be very much widened.
Several of the insurance offices grant annuities, but
none, I believe, in which the annuitants are sharers in
the profits.

The details of a Friendiy Society comprise every pos-
sible species of life contingency. They grant weekly
payments during sickness, annuities in old age, and sums
payable at death, in consideration of weekly premiums.
These institutions, combined with Savings’ Banks, and
aided by the removal of the abuses of the Poor Law,
will, in time, raise the labouring classes of this country
to a degree of independence which they have never
known. But, as might have been expected, the manage-
ment of these important institutions has, in many in-
stances, been wanting in prudence ; and I am afraid it
is hopeless to expect that the unity of system, which

u 4

ra]

2900 ESSAY ON PROBABILITIES.

must prevail before a thorough knowledge of the ad-
vantages they offer can get abroad, can be attained while
their several administrations are unconnected, and at
liberty to pursue all possible variety of plans, subject
only to the certificate of an actuary that each propo-
sition is not unsafe. But something more than safety is
required: an equitable distribution of benefits, and a
certainty of the most careful management, are as neces-
sary to the universal formation of these societies as an
opinion of their safety. The government, which
has within these few years been compelled, by the
most decided necessity, to apply a very severe and
searching remedy to an abuse of long standing, owes
the labouring classes a strong expression of sympathy
with the numerous cases of hardship which such a mea-
sure must create, and with the excellent conduct and
temper under its operation which has pervaded the
classes most immediately affected by it. It is to be re-
gretted that the change itself was not accompanied by
acts of parliament for the encouragement and aid of
societies such as those of which I am now speaking, in
addition to those which already existed for their regu-
lation. The most determined opponent of the protec-
tive principle would hardly dispute the policy of giving
effective help to the efforts of self-support, at the mo-
ment when the aid of the parish, which had been the
resource against poverty, became only the last security
against starvation. If the nation had been obliged to
abandon a distant colony, in circumstances of danger
and distress, there is no doubt that the settlers would

have been furnished with arms, arsenals, ships, money,

and all that could enable them to do whatever might be
done for their own defence and support. Has similar
help in similar circumstances been given at home? Is
the labouring man, thus suddenly thrown into a position
where the power and the habit of depending on himself
are necessary to a degree of which his training never
implied the existence, one bit nearer to the acquisition
of the power or the formation of the habit, by any aid

MISCELLANEOUS REMARKS. 297

of the legislature? Have even the opponents of the
measure, with their professions of benevolence, ever
pressed, or even suggested, the duty of showing the
labouring man, not only that by combination his class
can provide for itself, but that the community which
found it necessary to make a change involving him in
years of uncertainty and possible hardship, was desirous
that he should have that knowledge, and willing to aid
him in attaining its full benefits ?

It is not too late to take the necessary steps; and any
one who imagines a legislature able to feel, or to think,
will see the means of addressing himself to the first
faculty by such considerations as the preceding, and to
the second by urging the policy of giving every class a
share in the artificial system of property on which the
country now depends. At present, the property of a
labouring man is all tangible, and immediately at hand;
it would not be a great wonder if he were found to
have no clear opinion of the rights of a landlord, a fund-
holder, a mortgagee, or an annuitant. But if he him-
self were in possession of any of those claims which, by
means of law, can be created, enforced, or transferred
by virtue of the possession of a bit of paper— still more,
if the support of his old age and of his sick bed were con-
nected with this purely legal tenure of his past savings,
he would then be interested in the preservation of the ex-
isting system by the share of it which belongs to himself.

The friendly societies, numerous as they are, are by
no means universally distributed ; and if they were, the
smallness of their several amounts of investment must
occasion the expenses of management to bear a larger
proportion to the whole than would be the case if all
were united. Besides which, it happens every now and
then that the affairs of such a society fall into disorder
from want of skill or care. The government has lent
considerable assistance by allowing their investments a
larger rate of interest than could elsewhere be obtained ;
but this aid, independently of its being but little known
_ by the class whom it most concerns, does not guarantee

298 ESSAY ON PROBABILITIES.

the proper use of the funds so invested. If one large
office were to be established in London, having the
general management of the money raised, and the regu-
lation of its distribution, it would not be difficult to find
persons of station * and character throughout the country
who would consent to act as agents, receiving the con-
tributions and certifying the claims. The expense of
management might be borne for a few years by the
public purse, and this burden might be gradually thrown
on the establishment itself. No very great difficulties
could arise in the formation of such an institution, and

certainly none the expense of conquering which would

not be trifling in comparison of the greatness of the object
gained. The act which should establish this universal
Friendly Society would, in two generations, become the
real poor law.

The subjects of fire and of marine insurance are
founded on principles of great simplicity, though it is
not easy to procure exact data for the computation of
risks. As there exist no offices which are managed on
the republican method of a mutual Life Insurance Com-
pany, no publication of the results of experience has
been made. If every loss by fire or sea were a total
loss, it would only be necessary to ask what proportion
of all the houses or ships now existing is burnt or
wrecked in a year or on a voyage, and the premium for
insuring a house for one year, or a ship for one voyage,
would immediately follow. Thus, if of all the ships
which sail to the West Indies, one in a hundred is lost,
the lowest premium at which an insurance could take
place is one per cent., and all demanded above that pro-
portion would be profit. It would not, perhaps, be
very easy to ascertain this proportion with exactness,
and the difficulty is increased if ships or houses be
divided into different classes as to security, since the
tisks of each class must be ascertained separately. _ But

* Many of the Friendly Societies now established depend almost entirely
upon the superintendence of the clergy or local gentry.

MISCELLANEOUS REMARKS. 299

the greatest obstacle to a satisfactory adjustment of
risks, lies in the necessity of taking into account the
chances of only partial loss, which would make the
tables (if they could be procured) nearly as complicated
as life tables.

On the subject of marine insurance, nothing is
known to the public, as to the experience of the
underwriters; and, as it is not directly interested
in the subject, it would be difficult to create any
disposition to inquiry. The mercantile world, how-
ever, and the underwriters themselves, have a direct
interest in the dissemination of such information,
for reasons which it is no pleasant task to state,
both on account of their invidious character, and their
obvious want of connexion with the general objects of
this treatise. But the latter circumstance may, perhaps,
not be disadvantageous, since the statement of the exist-
ence of an imputation, coming from a quarter in which
there is no interest whatever, either in the continuance
or discontinuance of any present condition of things;
need not excite any disposition, except that of calmly
weighing whether it is necessary or not to produce a
refutation.

Some years ago, I heard the following opinion stated
in a mixed comparty, in reference to a then proposed
attempt to render ships incapable of actually sinking,
however much they and. their cargo might be damaged ;
namely, that the mercantile world would not be inclined
to patronise an invention which would make the seaman
safer than the ship. Some time afterwards, I saw an
article in a periodical journal, distinctly written for the
purpose of making its readers believe that, in conse-
quence of insurance, unsafe ships are allowed to be used,
to an extent which has caused much more loss of life
and property than could have been experienced if no
such institution had existed. Other allusions, more or
less direct, in various publications, have convinced me
that one of two things must be true, either such an
impression has a party who acknowledge it, or authentic

300 ESSAY ON PROBABILITIES.

information upon the subject is so difficult to be ob.
tained, that the one, two, or ten, who believe it, or
profess to believe it, feel that no answer can be made to
the assertion.

That there are men in the carrying world (if mer-
cantile world be too wide a phrase for the subject) who
would, from a pitiful economy, expose the seaman to
risks which a little outlay might prevent, is very pos-
sible; there are men of such a spirit in every world:
that there are others who would consider such conducé
as little short of murder, a like analogy would equally
justify us in asserting. Which class has predominated
can only be absolutely known to the public by results,
without which there is but general opinion upon cha-
racter to aid any individual in forming his conclusion.

. It is in human nature that the insured should not be so

careful as one who stands risk ; and it is, unfortunately,
the general experience of men acting in bodies, that they
are not found to be swayed by the principles which
would be acknowledged and acted upon by them severally.
Putting these things together, it is not wonderful that,
in any case where suspicion might attach to a body of
men, there should be quarters in which it does attach.
It would not be wonderful, either, if the suspicion were
found to be perfectly groundless ; but correct feeling
would point out the desirableness of forestalling such
suspicion, if possible, by the publication of all necessary
information. In the present instance, it would be well
that the proportion of loss, among insured vessels, should
be known ; it would not be necessary to state the values
of the several vessels, since the simple account of the
number insured, and the number on which a claim has
been paid, in various years, would be sufficient. The onus
of proving that the loss on uninsured vessels, or on
vessels which sailed before insurance was known, is or
was greater than that on insured vessels, would-lie upon
those who make the charge. All persons, in the case of
any body of men, must hold every thing short of ab-
solute proof against them to count for nothing, when

MISCELLANEOUS REMARKS. 301

_they show themselves ready to communicate those ma-
terials out of which a misdemeanor, if there be one,
might be substantiated.

The offices for the insurance of fire have not given
any account of the proportion of insured houses upon
which claims have arisen. ‘Their usual annual charge
is, I believe, about one part in a thousand of the sum
insured, upon premises of ordinary risk, such as a
dwelling-house in London. There are higher rates
for more hazardous insurances, constructed, I should _
imagine, very much from mere estimation of the risk.
But the government steps in between the insurer and the
insured, and imposes a duty on each policy which
nearly trebles the annual payment upon it. This has
been called a tax upon prudence, and in like manner
the stamp duty might be called a tax upon justice. I
am afraid that if nothing commendable suffered under
an impost, reformation would thrive more than re-
venue ; and a deficiency of means to pay the interest
of the debt would be a heavier tax on prudence, justice,
and every thing else, than any minister has yet con-
templated. But it may not therefore follow that the
particular tax in question is politic, still less that its
amount is justifiable. The reason of the tax is plainly
this: the moral security offered by the fire office is
worth so much more than competition will allow them
to ask, that the impost is one which does not fall so
heavily as it would do if levied in many other quarters,
Nobody can question the truth of this; but, never-
theless, the amount of the tax imposed by the legis-
lature must be owned to be excessive, and likely to act
as a prohibition in the case of poor persons occupying
small premises.

But there is a mode of overthrowing this tax, or, at
least, of bringing the government to terms, to which I
can see no impediment, practical or moral. It is the
application of the principle of mutual insurance by a
number of individuals acting in a private capacity, and
not opening a public office. Suppose a thousand indi-

302 ESSAY ON PROBABILITIES.

viduals, registering their names, to appoint three men of
undoubted character to receive contributions of one
guinea a year each. If the subscribers be occupiers of
dwelling-housesin London, there is no doubt that this sum
would be amply sufficient to insure a thousand guineas
to each. If three years were to elapse without a fire
taking place, the subscription might be suspended, until
circumstances should diminish the fund; which, im-
proving in the mean time at interest, would become
every year more capable of meeting demands upon it.
‘There would be no need of any legal security, if the
trustees were well chosen ; and a short agreement would
explain the understanding on which the parties contri-
bute. As soon as a few such clubs were formed, the
inutility of imposing a tax on one particular way of
effecting an object would become apparent.

It would be lucky for the preceding plan, if it were
the decided opinion of lawyers that the courts of equity
would not entertain any application for inquiry into the
state or management of such funds; since, in that
ease, the law of honour would be sufficient. It has
always been found, that whenever the law of the land
refuses to protect a proceeding which is fair and equal
in itself, a stronger law claims jurisdiction. The parties
benefited in the end would be the fire offices, since such
a method of resisting this excessive tax would inevitably
procure its abolition.

There is another tax which, though not so dispro-
portionate in its amount, is much worse in its principle
than that on policies of fire insurance : namely, the tax
on policies of life insurance. It must be remembered,
that the income of which the savings are invested in
this manner, has already undergone a considerable
amount of taxation, If any investment of such savings
be taxed, all should be treated alike.

The abolition of lotteries happily leaves nothing to
be said upon the subject of gambling, encouraged and
promoted by the government ; and the recent decision
of the French legislature, by. which the public gaming-

MISCELLANEOUS REMARKS, 303

houses have been suppressed, must be a source of con-
gratulation, both from the excellence of the measure
itself, and the prospect of imitation which it opens, on
the part of other continental powers. But, at the same
time, it cannot be denied that, however desirable it may
be that no community should give to gambling that
appearance of sanction which is implied in regulation,
the refusal of the latter is accompanied by evils, of
which it is never possible to say positively that they fall
short of those which would be produced by sufferance -
accompanied by restriction. In this country, there are
the means of gambling open to every class of the com-
munity, and there can be no doubt that those who avail
themselves of them are subject to imposition in a degree
which could not be the case if the play were accompa-
nied by publicity. The classes of rank and wealth
have the power of forming themselves into clubs,
in which illegal games are played without the possibility
of detection, and .in such institutions there can be no
doubt, with rare and occasional instances of exception,
the play is conducted at least with fairness. But no
such thing can be supposed with regard to the numerous
receptacles in London and other large towns, and which
are believed to exist in different forms, suited to all
classes of society. The difficulty of obtaining legal
proof renders conviction next to impossible ; and the
occurrences which sometimes take place at the sessions,
prove that, even when enough of evidence is obtained to
hold parties to bail, the accused can generally find the
means of preventing the evidence from being forthcom-
ing to sustain the indictment. Under such circum-
stances, gambling in its worst form thrives in defiance
of law. Nevertheless, the good consequences of dis-
couragement are visible throughout the country. There
is no people in the world among whom so little of direct
gambling is found.

The infatuation which leads persons to suppose that
they can ultimately win from a hank, which has chosen
a game in which the chances are against the player, is

304 ESSAY ON PROBABILITIES,

one which can only be cured, if at all, by a quiet study
of the theory of probabilities, Perhaps some of our
readers may suppose, that the persons who thus court
ruin, do it under the notion that the results given by
that theory are dubious, or derived from unpractical
speculation, or perhaps absolutely false. So far is this
from being the case, that though they undoubtedly fall
into error by forming their notions from observation
unaided by theory, yet their error frequently consists
in representing games of chance as being more unfa-
vourable to themselves than they really are. Though
the true premises should lead them to the conclusion
that success is next to impossible, they cannot learn the
truth even from a mistake which should teacn it @
fortiori. The author of the article “Gamine,” in the
Penny Cyclopedia, states, apparently from his own
knowledge, that it is customary to consider the chances
of the bank at the game of rouge-et-noir, as 7% per
cent. above those of the player. Now, it can be imme-
diately shown, from the first appendix, that when a
player puts down a stake, his chances of doubling his
stake, of losing it, and of simply recovering it, are as
8903, 9122, and 1975. Now 9122 does not exceed
8903 by 73 per cent. of 8903, but only by about 23 per
cent. If, however, the preceding assertion meant that
the game was considered as a simple one, in which the
chances were as 46} to 53%, the error was very large

indeed. So far as this one instance goes, it should

seem that the warning against this game, as derived
from observation of its results, was yet stronger
than that which would have been given by the
theory of the game. The same author adds, that he
heard it frequently asserted by constant frequenters
of the Parisian gaming-houses, that it was absolutely
impossibie for any one to win in the long run.

Still, however, to the hopeless attempt of squaring
the circle, or of finding perpetual motion, we have to
add that of discovering a method of certainly winning
at play: the attempt at which has been the ruin

Es
S
=
:
4
F

MISCELLANEOUS REMARKS, S05

of many a speculator. The gaming banks have dis-
covered the secret, which is simply to embark consider-
able capital, and to play with chances unequally in
their favour. To produce in the young mind a con-
viction that events will happen, in the long run, in a
fixed, and not in what is called a fortuitous manner,
should be an object of education, in order to produce
that soundness of views on the results of gambling
which is a sure protection against the temptation. By
trying experiments upon what are called chance events,
such as might easily be done with a pack of cards, or
a few dice, it might easily be made to appear that no
large number of events will present any marked devi-
ations from the general average which the knowledge
of this theory points out before-hand. Persons aware
of the truth of the law just stated, may often be able
to apply it advantageously. I received the following
anecdote from a distinguished naval officer, who was
once employed to ‘bring home a cargo of dollars. At
the end of the voyage it was discovered that one of the
boxes which contained them had been forced; and on
making further search, a large bag of dollars was dis-
covered in the possession of some one on board. The
coins in the different boxes were a mixture of all man-
ner of dates and sovereigns; and it occurred to the
commander, that if the contents of the boxes were
sorted, a comparison of the proportions of the different
sorts in the bag with those in the box which had been
opened, would be strong presumptive evidence one way
or the other. This comparison was accordingly made,
and the agreement between the distribution of the se-
veral coins in the bag and those in the box, was such
as to leave na doubt as to the former having formed a
part of the latter.

306 ESSAY ON PROBABILITIES.

ADDITION TO CHAPTER X.

The following formule will sometimes be found
useful.

The present value of an insurance, which is to be
#1 if the party die in the first vear, £2 if in the second,
and so on, is —

1+A—rI
l+r
where A is the value of a common annuity ; I that of
an annuity which is £1 at the first payment, £2 at
the second, and so on; and ¢ the interest of €1 for
one year. 3

If an office engage to pay £1 at the death of an
individual, and also to return all the premiums at the
same time, that is, if they guarantee that the interest of
his investments shall amount to €1, the premium which
‘should be demanded is

K-A
1+A¢I
where A and I are as before, and E is the value of a
perpetuity of €1.

APPENDIX.

APPENDIX THE FIRST.

ON THE ULTIMATE CHANCES OF GAIN OR LOSS AT PLAY,
WITH A PARTICULAR APPLICATION TO THE GAMF
OF ROUGE ET NOIR.

Tuoucu the first part of the following reasoning is
of a mathematical character, I have been induced to
insert it by the consideration that the results of page 109.
have never yet been introduced into an elementary
work, nor even proved to the mathematician except either
by incomplete or complicated trains of reasoning. Such
being the case, perhaps even a well-informed mathe-
matician might be excused for doubting some of the
results of chapter V., and I have therefore digested the
following demonstration, that no one who bears such a
character may be able to weaken the evidence for the
necessity of the pernicious results of gambling which
that chapter is intended to afford.

De Moivre was the first who gave a solution of the
following problem, and by a method of the most striking
ingenuity. But his demonstration has the defect of
assuming that one or other of the players must be
ruined in the long run. Laplace* and Ampére,—the

* The solution of Laplace gives results for the most part in precisely
the same form as tho-e of De Moivre, but, according to Laplace’s usual
custom, no predecessor is mentioned. Though generally aware that La-
place (and too many others, particularly among French writers) was much
given to this unworthy species of suppression, I had not any idea of the
extent to which it was carried until I compared his solution of the problem
of the duration of play, with that of De Moivre. Having been instru-

x 2

ij APPENDIX THE FIRST.

former in his Théorie, &c., the latter in a tract entitled,
Considérations sur la Théorie Mathématique du Jeu,
Lyons, 1802.—have also solved the problem: both so-
lutions are of the highest order of difficulty, and cannot
be rendered elementary. If my memory be correct, I
have seen references to other solutions.

The problem is as follows :— Two players, A and B,
the first possessed of m times and the second of n times
his stake, play at a game so constituted that it is a@ to }
that A shall win any one game; required the proba-
bility which each has of ruining the other, if the game
be indefinitely continued.

I shall first take the case where one of the players, A,
is possessed of unlimited means, that is, in which m is
infinite. Let By,» represent the probability that
B having n counters, shall ruin A who has m counters.
Then, if m be infinite, B,, .. will after the first game,
become either Br+i, 0, Or By, _1, , according as
that game is B’s or A’s; of which the chances are

b a
and. i...
a+b a+b
Consequently,
b
Big i Pt 0 Py Bee ee

which gives
ay
Brio =C eee
which, when C’ and C” are determined, will represent

the probability that B will never be ruined, but will
continually gain more and more from A. But the

mental (in my mathematical treatise on Probabilities, in the Encyclopedia
Metropolitana) in attributing to Laplace more than his due, having been
misled by the suppressions aforesaid, I feel bound to take this opportunity
of requesting any reader of that article to consider every thing there given
to Laplace as meaning simply that it is tobe found in his work, in which,
as in the Mécanique Céleste, there is enough originating from himself to
make any reader wonder that one who could so well afford to state what
he had taken from others, should have set an example so dangerous to his
own claims.

ULTIMATE RESULTS OF PLAY. iil

same equation (1) is equally true if for B,, .., we substitute
either 1—B,., or Ax, n, which two last may be
different, for any thing yet proved to the contrary. In
fact, the equation (1) is merely the general expression
of the condition that n is changed ‘into n+ 1 or n— 1
according as one or the other of two events happens,

a
whose chances are aa and rigs It may be seen
however, immediately, that in the case of n= 0, in
which case the proposed contingency becomes an initial
impossibility, we must have B 5, ..==0, or C’+C” =0.

We have then
B,,, o= Org {1 -~ (;-)" }

This result is rational only when a is not greater
than b, unless we suppose C’=0. But the necessity
for investigating what takes place in this case is saved
by observing a very simple relation which exists between
Bx, m, and B,~, Supposing A to have infinite means,
it makes no difference in the state of the question if we
take any number of stakes m from the stock of A, and
suppose that they shall be lost before the rest are touched.
Consequently B cannot win indefinitely from A unless
he first ruin A’s stock of m stakes, and afterwards, be-
ginning from n+m stakes, win indefinitely from A’s
‘emainder. That is

NM, 0 nm, m iu + M0
or B, mi B,, 0 Br +m, -
or {1 ds #co"f1— Gs Vester
ered Ge ia

na+m m+n
b —a

Whence, applying the same reasoning to A m,n, we
find that if two players A and B, possessing m and n
xs

iv APPENDIX THE FIRsT,

stakes, play at a game which gives a to b in favour of A |
at each trial, then, in an indefinite number of trials

Nook Go age cae .

B,,, mM, the chance that B shall ruin A= wnt “a r ere!
- a” ha _ Py )
A the chance that A shall ruin B=
mM, Ny 4” pan ay m a lie

The sum of these two chances is unity, from which it
appears that one or other must be ruined in the long run.
These results agree entirely with those of De Moivre,
and all the rules in page 109. may be easily deduced
from them.

It appears also, that if the conditions of any game,
however complicated, can be reduced, in the case where
one of the players (A) has unlimited means, to the
equation

Baio =8 Bn +1, 0, + % By — 1,0 (where a+B=1);

then the ultimate results of that game exhibit probabi-
lities of precisely the same value as those of a simple
game, in which it is a to 6 for A against B.

If, besides cases in which A or B wins, there be
cases in which the game is drawn, no alteration is made
in the result (though the number of games in which
there is a given probability of either party winning
must of course be increased.) Let a, (2, and 0, be the
chances that A wins, that B wins, and that the game is
drawn: then (a+$+6=1)

Brio =8 Bn + 1,0 +4 Br—1,0 +5 Bn, x or

B a B a
Bn, « = Bn +1,0 + 7 5Bn-1,0 — + >4=!)

the same as in a game which must be won or lost, and
in which it is a to @ for A against B.

The following is the problem of the game of rouge et
noir, which I shall afterwards proceed to explain.

ULTIMATE RESULTS OF PLAY. Vv

A and B play at a game which presents four cases,
A, B, D, and T, of which the chances are a, /3, 6, and 0,
so that a+ (3+6+0=1. When A happens, the player
A wins ; when B happens, the player B wins; when D
happens, the game is unconditionally drawn ; but when
T happens, the game is drawn, and in the next game
only the player A puts down a stake, and not the player
B. If D should follow T, or if T should happen any
number of times running, or D, or successions of T and
D, still A’s stake remains risked, without any from B:
nor does B stake again, until the happening of A or B
recovers A’s stake, or assigns it to B: after which, both
parties stake again.

Supposing the means of B to be unlimited, let A,,
represent A’s chance of winning indefinitely, immediately
before a game in which both are to stake, A having m
stakes in his possession: and let A’,, represent A’s chance
immediately before a game in which B does noé stake.
Then, by the preceding method

Am =%Am 41+BAm—1 + 5Am + 0A'm
Eliminate A’,, and we have

_ _a(a+8) 4 B(a + B) + 0B
m~ a+ pron ™+1T (4624 Op Am—1

in which the sum of the two co-efficients is unity.
Hence this game is equivalent in its ultimate chances to
a simple game in which it isa (a+) tof (a+ B)+6f
for A against B. If a=, the last odds are those of
2a to 2a4+8.

This game of rouge et noir is described in an unin-
telligible manner, and with material omissions, in the
later editions of Hoyle, from which work, and from the
testimony of persons who have seen it played, I give the
best description I can make of it, observing that the
most modern method of playing differs in several parti-
culars from that given in the book referred to.

A number of packs of cards is taken (six, it is said in

x 4

vi APPENDIX THE FIRST.

Hoyle) and all the cards are well mixed. Each common
card counts for the number of spots on it, and the court
cards are each reckoned as ten. A table is divided into
two compartments, one called rouge, the other noir, and
a player stakes his money in which he pleases. The
proprietor of the bank, who risks against all comers, then
lays down cards in one compartment until the number of
spots exceed 30; as soon as this has happened, he
proceeds in the same way with the other compartment.
The number of spots in each compartment is then be-
tween 31 and 40, both inclusive, and that compartment
wins which has the lower number of spots; so that if,
for instance, there should be 37 spots in the rouge, and
32 in the noir, those players who staked upon noir
would win from the bank sums equal to their stakes. If
the number of spots be the same in both (which is called
in Hoyle a refait) the game is drawn, and the parties
withdraw, diminish, or augment their stakes at pleasure,
for a new game: except only when the number of spots
in both compartments is 31 (called in Hoyle a refait
trente et un), in which case the bank is allowed to with-
draw its stakes, and those of the players, whatever their
compartment may be, are impounded (placed en prison).
In the next game (now called an aprés), the impounded
stakes are played for, the players choosing their com-
partments as before: should the bank win it takes the
stake, should the bank lose the player recovers his stake.
Should a second refait trente et un occur, or a drawn
game, the stakes still remain impounded, and are not
released until a gain or loss arrives. In the meanwhile
new stakes may be put down, before the fate of the old
ones is decided. ;

The chances of this game depend, in a slight degree,
upon the number of packs of cards which are mixed
together. When, however, there are as many packs as
six, it is very nearly * indeed the same thing as if the

* The only ways in which 31, for example, could be obtained from an un-
limited number of packs, and which could not equally well be obtained from
six packs, are those in which more than 24 aces occur. Now the proba-

ULTIMATE RESULTS OF PLAY. Vil

number of packs were unlimited. The following tables,
computed upon the latter supposition, will represent, with
more than sufficient accuracy, the chances of the several
arrivals. The first three columns of the first table ex-
hibit the chance of arrival of each number among the
several sum-totals which precede the arrival of thirty-one
or more. ‘Thus opposite to 4 we see ‘0961, which is
the chance that the beginning of the sequence drawn
shall show one of the following sets of cards (1 standing
for ace, &c.):

,1,1

2 2,2
1, 3, 4,
3,1

1 | 0769 11 1200 § 21 "1398 31 148]
2 | °0828 12 | °1247 22 | °1424 82 |: :ta72
3 | °0892 13 | °1293 23. | °1449 $3 |..:4275
4} 0961 14 | °1340 24 | °1472 $4 | *1169
5 | °1035 15-1 “1386 25 | °1493 35 | *1060
oe 7 114 16: | °1432 26 | ‘1511 36 | °0956
7 | °1200 17 | °1476 27.1 | <R52T 387 | °0838
8 | °1292 £8...\..21 519 28. | *3541 38 | :0723
9 | °1392 19 |.°1559 29 | °1552 | 39 | °0607
10 | *3806 20 | °2129 30 | °1683 40 | °0518
3l | °0219 | 34 | :0137 | 37 | -0070 40 "0027
32 | -0190 | 35 | °0112 § 38 | :0052
33 | 0163 | 36 | 0090 | 39 | :0037 | D. G. ‘0878

The fourth column contains all the chances of those
points which lie between 31 and 40, that is to say, the
chance of each being the first which arrives. Thus
1060 is the chance that 35 will appear, and that it will
be the first which appears above 30. The second table
(containing the squares of the numbers in the last column
of the first) shows the chance of each refait, that of the

bility that out of 31 cards drawn at hazard, 24 or more shall be aces, is
altogether beneath consideration, It is less than one out of a million of
million of millions.

Vili APPENDIX THE FIRST.

refait* trente et un being ‘0219. Opposite to D. G. is ©
the sum of the chances of all the refwits except the first, —
which sum is the chance of a drawn game.

If then we say that °021 is the chance of a refuit
trente et un and ‘087 that of a drawn game, there remains
*892 for the chance that either the bank or the player
must win; which chances being equal, give °446 for
the player, and the same for the bank (exclusive of the
benefit of the aprés). Returning then to the result in
page v., we find a=(=—°446, 0 =:021, and 2a to 2a
+6 is ‘892 to *913, or 892 to 913. Consequently ; —

At the game of rouge et noir, as now played, the
chances of ultimate ruin to the bank or the player are
the same as they would be at a simple game, which
must be either won or lost at each throw, and in which
the bank has 913 chances of winning, where the player
has 892.

The bank, as noticed in page 110., is playing against
the whole public, or against a player with unlimited
means. Taking 892 to 913, or more correctly 8903
to 9122, and applying the rule in page 110., the follow-
ing table results, which must be thus used. Opposite
to 30 we find °4824, which is the chance of the ultimate
ruin of a bank which risks one-thirtieth of its means at
every game: —

10 "7843 110 “0690 210 0061
20 6151 120 0541 220 0048
30 *4824 130 0425 230 0037
40 3783 140 0333 240 0029
50 "2967 150 0261 250 0023
60 2327 160 "0205 260 0018
70 "1825 170 0161 270 0014
80 1431 180 0126 280 ‘0011
90 "1122 190 “0099 300 “0007
100 ‘0880 200 0077 400 0001

* The editor of Hoyle says, or implies, that the chance of the arrival of
$1 is one-eighth, or °125. This is, no doubt, a conclusion drawn from ob-
servation. The table in Hoyle, exhibiting the odds (page 147., which refers
to 141:, edition of 1814), is altogether erroneous, -

ULTIMATE RESULTS OF PLAY. ix

Hitherto all our results seem in favour of the bank ;
that is, tending to show that its advantage is not so
great as is commonly supposed. No person, granting
a bank permission to exist, would grudge it such an
advantage as would make it 49 to 1 against its being
ruined by the possible fluctuation attendant upon an
unlimited duration of play. This chance of being
ruined, namely ‘02, appears from the table to be ex-
ceeded, unless the bank possess 160 times the sum
risked at each game: if this were 100/., the bank would
need a capital of 16,0007. But I must now request
attention to the other side of the question; first, con-
sidering the bank against the public; and, secondly, the
bank against an individual player. One of the most
important features in this game (which springs from
the old game of Faro, as did the last from the still older
game of Basset*) is, that the bank does not risk the
whole sum it lays down, but only the difference between
those sums which the caprice of the players obliges it
to stake on rouge and on noir. If 20 players have each
staked a guinea, 12 on rouge and 8 on noir, and if rouge
win, the bank loses 12 guineas and gains 8, and conse-
quently did not risk more than four guineas. It is im-
possible to say what chance there is of the bank having
to risk a given sum in such a case, as this depends on
the will of the players. When the cards have several
times decided for rouge, those players who think the
run is not finished will stake on that colour, while others
who think differently will stake on noir. I am wholly
without the means of saying what average exists, but I
should incline to think it very unlikely that the bank
really risks more than one fourth of its deposits.t But
the advantage which it derives from the refait trente et

* An assertion of the editor of Hoyle, which is true as to the principle of
the game — namely, that besides equal chances for the bank and the
layer, there are chances for a drawn game, and a case in which the bank
oe a direct advantage amounting to half the stake — but the details are
very different. Both games are described in De Moivre.
+ The bank is evidently (its chance of the aprés excepted) merely the
means of equalising the sums staked on the two colours.

x APPENDIX THE FIRST.

un, and its consequences, is gained on the whole of the —
opponent stakes. The following is the method of esti- —
mating the mathematical advantage of the bank: —Before ~
a common game, the prospects of the bank lie entirely ©
in the chance of that game being followed by an aprés,
‘since, in all other respects, the chances of gain and loss —
are the same. After a refait trente et un, on whichever |
colour any player may ‘choose to risk his impounded |
stake, the bank has the chance a of winning that stake, —
and none of losing. But besides this, the bank has all
the chances of a second refait (or 0+6 or 1—2a), for
having another trial of the same kind: if then # express
the fraction of a pound, which such a chance of 1/. is
worth, we have

w=a + (1—2a) x, or v=},

The mathematical advantage of the bank is therefore,
the chance @, or ‘0219, of being put in possession of the
worth of half the stakes; or ‘O11/. of all the sums
it deposits. This is 11. 2s. per cent.* per deposit ;
which, to those who know the rapidity with which the
risks succeed one another, will appear to yield, in the
course of the year, an ample return, not merely to the
deposits, but to the sums which are reserved for security
against fluctuation. Itis probably 45 per cent.upon every
real risk ; and the return in the course of a year may be
easily guessed at. Imagine 100 different games, played
on each of i100 different evenings, the sum risked by the
collective players on each game being 50/. The total de-
posits of the bank would be 500,000/., on which 1/. 2s.
per cent. is 5500/. The capital required to make this spe-
culation much more safe than any mercantile adventure,
would not be iarger than its probable return in one year.

* Some time after this was written I chanced to find the following
sentence in the lately published Theory of Probabilities of M. Poisson.
** Dans les jeux publics de Paris l’avantage & chaque coup est peu considér-
able: au jeu de trente-et-quarante par exemple, il est un peu au-dessous
de ‘011 de chaque mise. Voyez sur les chances de ce jeu, le mémoire que
j’ai inséré dans le journal de M. Gergonne, tome xvi. numéro 6, Decembre,
1825.”” 1 have not seen this memoir ; the accordance of the result with my

own, shows that I have described the game correctly, as it was played in
Paris ; of which, from paucity of information, I was by no means sure.

ULTIMATE RESULTS OF PLAY. xi
If any persons, aware that the preceding calculations
are new, should imagine that there must be some mis
- calculation in a result which shows that cent. per cent.
on the necessary capital might be gained three times in
a year, I reply, that the chance of a refait trente et
un, as given by the editor of Hoyle, produces a result
of nearly as surprising a character. For ‘0219 read
0156, or ,1,, and the 5500/. above-mentioned becomes
a trifle less than 40002 The preceding results,
or either of them, being admitted, it might be sup-
posed hardly necessary to dwell upon the ruin which
must necessarily result to individual players against
a bank which has so strong a chance of success
against its united antagonists. But so strangely are
opinions formed upon this subject, that it is not
uncommon to find persons who think they are in
possession of a specific by which they must infal-
libly win. The last table given. will show the
chances which any single player has of ruining the
bank, and of being ruined himself, as follows: —If
the player stake one mth part of his means * at each
throw, and the bank one nth part, from unity subtract
the number in the table opposite to m and to n+ m,
and the first result divided by the second shows the
chance that the bank will ruin the player. Suppose,
for exaniple, that the player risks 1-10th and the
bank 1-160th of their respective resources. Then
opposite to 160 and 160 + 10, we find ‘0205 and °0161 ;
which, being subtracted from 1, give °9795 and -9839,
whence $432 is the chance for the bank ; or it is 9795
to 44, or 223 to 1, that such a player will be ruined.
Even if both the player and the bank stake 1-160th
parts of their several funds, the bank will still have -98
or 49 to 1 in its favour against that one player.
The last column of the table in page vii, shows that
the bankers at rouge et noir, by making the aprés de-

* We must not here consider what the bank stakes against the individual
player, but the whole sum which it risks.

Xii APPENDIX THE FIRST.

pend on the refait trente et un have chosen the most
favourable out of the ten cases. The following table will
show the effect of substituting any other refait ; the
first column pointing out the refait in question; the
second, the simple game to which it is equivalent in the
chances of ultimate ruin ; the third, the benefice of the
bank upon every 100/. deposited :—

es le
31 10,000 _ to 10,246 - tae ee
$2 © - 10,213 - OTs. @
33 * - - 10,183 - 016 6
34 - - 10,154 - O-ts: &
35 - - - 10,126 - O11 O
36 - - ree its Gee 0.2 oe
$7 - ~ - 10,079 - at eae ee
38 - - - 10,058 - GO Fm
$9 - “ - 10,042 - 0 4 0
40 ~ - - 10,030 - O° S29

The above is a graduated scale of poisons, each one
being slower in its operation than the preceding ; the
first, or quickest of all, being that which is used at
present. Of all the illegal games, none that I know of
is less likely to lead to ruin than rouge et noir ; and, the
results of this investigation give a sufficient notion of the
state of the case between the banker and his dupes.

The first table, in page vii., is calculated in the fol-
lowing manner: — The chance of any given card at
a given point is 54, for every number which o card
can give, excepting 10, the chance of which is ;*;, on
account of the value given to the court cards. Let w be
a number greater than 10, and let V, be the chance of
arriving at that number in the laying down of the cards.
Then, if (V,) signify the event of which the chance
is V,, and if by (a) (b), we mean the consecutive
happening of the two events whose chances are a and 8,
it follows that (V,) when it happens, must happen in
one of the following ways: —-

ULTIMATE RESULTS OF PLAY. xiii

( Ve-1 ) (ts) or (Vz. ) Gs) - -- or (Vig ) (a3) oF
(Veto) ‘s)
or V,, = (Ve-1 + Vans toeet V )+8V

Z-9 r-10
from which it is readily found that

AV, =% (Vex ee Vig Vino. )
The first eleven values of V; are thus determined: V, is
evidently yg; V, is 7, V, + 75 OF 73 - 433 Vs isay V, +45 Ve

+ 7, or 7; (4$)2; and so on up to V,, which is

ty (V1 + Vot.. + V9) + or 5 (4) + %
Vi is 3 ( Vio VEE eS Vo) + 7 Vi or
fa (H)O# 6 (Gh)?

The rest were then calculated by the preceding for-
mula for AV, to six places of decimals (by which the
accuracy of four was insured) as far as V31 inclusive.
The remainder were calculated by those which preceded,
leaving out the terms which the necessary distinction
already mentioned requires to be omitted.

It may perhaps appear to some that a part of the
preceding reasoning is inapplicable, since it only calcu-
lates the chances of ruin in an indefinite succession of
games, whereas any practicable number of games, though
great, may not involve the same chances as an infinite
number. The objection is valid in principle, but the
correction which is rendered necessary by it is not worth
consideration, if any large number of games be in ques-
tion.

The following are the only cases in which a simple
approximate rule can be given, connected with finite
numbers of games : —

Prositem. Both parties have the same number of
stakes (which should not be less than 20), say a, and
the play is equal, or either has an even chance of win-
ning any one game. What is the chance that one or
other shall have been ruined before # games have been
played ? (# being a large number).

XiV APPENDIX THE FIRST.

Ruue. Divide 30 times w by 56 times the square of
a, and from the quotient subtract ‘1049. If the result }
be the common logarithm of z, then z — 1 to 1 are the
odds in favour of the event.

Examrte. The number of stakes is 45, and the
play equal: what is the chance that one or other is
ruined before 1520 games have been played?

a=45,7=1520, 56xaxa=113400

45600
zx x 30 = 45600, 113400> ‘4021

"4021 — *1049 = +2972 = logarithm of 1°982.
Answer, ‘982 to 1, or 982 to 1000,

Propiem. The stakes being equal, and also the play,
as before, what is the number of games in which it is n
to 1 that one party or the other will have been ruined ?

Rute. To the common logarithm of one more than
n add *1049: multiply the result by 56 times the square
of the number of stakes, and divide by 30, which gives
the number required, very nearly.

ExAmp te. Both parties have 50 stakes: in what num-
ber of games is it10 to1 that one or other will be ruined?

a=50,n=10,n+1=11, log. 11 = 1:0414

1-0414 + *1049 = 1'1463, 56 x a x a= 140000
160482

1:1463 x 140000 = 160482: ——3.— = 5349

Answer: in about 5349 games.

To find the number of games in which it is an even
chance that one or other will be ruined, from three-
fourths of the square of the number of stakes, subtract
its hundredth part. Thus, if both parties have 40
stakes, then 40 x 40 being 1600, three-fourths of which
is 1200, from 1200 subtract 12, which gives 1188 for
the number of games (very nearly) in which it is an
even chance that one or other will be ruined.

If a player with a stakes play with one of unlimited
means, the chances being the same for both, it is an
even chance that he is ruined in a number of games

VALUE OF COP YHOLDS. XV

which is thus found: take 23 of the square of a. If
greater accuracy be required, add to the result one less
than its 760th part, which is sure to make it correct
within a single game. Thus if the number of his stakes
be 100, 100 x 100 x 23 is 23750, the 760th part of
which is 31, whence 23750+ 30, or 23780, is within
one of the number of games required.

APPENDIX THE SECOND.

*

ON THE RULE FOR DETERMINING THE VALUE OF
SUCCESSIVE LIVES, AND OF COPYHOLD ESTATES.

Tue rule given in the work is in a different form from
that of any writer with whom I am acquainted, though
it agrees with that given by Mr. Milne, as will be shown.
This Appendix has been rendered necessary by the fact
that no writer has solved the question of the value of
copyhold estates with absolute correctness except Mr.
Milne, whose solution is in a form of unnecessary diffi-
culty. The writers with whom I am acquainted, who
give the old rule, or one involving an omission of the
same kind, are De Moivre, Dodson, Thomas Simpson,
Stonehouse, Morgan, Baily, and the French translator
of the latter. Mr. Milne stands alone in proposing a
somewhat different rule, which like many results of
independent investigation, differs more than need have
been the case from the form of preceding results.

Let there be an estate held on a single life, and
renewable for ever upon payment of a fine of 1/7. Let
it be a condition, that each renewal is to be made on the
Ist of January next following the extinction of the
previous life ; it is required to find the present value of
all the fines. :

Firstly, To find the value on the Ist of January, the.
moment after a fine has been received, and the best life
which can be found has been put in. Let P be the

¥

XVi APPENDIX THE SECOND.

value of the life, or the value of an annuity upon it;
the rate of interest per pound, and F the value of all the
fines. Then upon the new year’s day next following
the extinction of that life, the whole value of the fines
will be 1+ F, because the person who claims the fines
will have one pound to receive, and will then have
remaining the interest which we have called F. It fol-
lows therefore that F is the present value of 1+ F to be
received at the end of the year in which the life drops:
or, by the well known formula

1+F F=—_—
5 As ) or a

where E is e the value of a perpetuity of 12.

Secondly, Let the life already in possession be of the
value A, the lives at each renewal having the value P,
as before. Consequently, the present value of the fines
is that of 1+ F to be received at the end of the year in
which the present life drops: and this is

1 oA E—A 1+E E-A.
ee: i ates Rts 1th the
If an estate be held on any number of lives, with a
fine of 1J. on renewing each, it is precisely the same
thing as if a similar number of estates were held on single
lives, and the present value of all the fines, the present
lives being worth A, B, C, &c., n in number, is

nE—(A+B+Cr....)
1+P

tne common rule divides by P instead of 1+ P.

A mathematical rule, when erroneous, is best exposed
in its extreme cases; let us then suppose the life P
certain to drop in the year. that is, worth nothing.
There will consequently be a fine to pay every ]st of
January, and the present value of what I called F in

the preceding investigation, is or E. The rule I have
given, shows that F=E when P =0; that of De Moivre,

VALUE OF COPYHOLDS. XV

Simpson, &c., makes F infinite. Again, suppose the

life P certain to last one year and to drop in the second ;

in which case its value is —— ioe Bere and F is a

perpetuity of 1/. receivable at the co marge of every two
2

If A= P=——., the new formula

years, or 2E+1° a 9 1’

becomes the old one becomes E, the value of

Fe
2h4+V
a perpetuity of 1/7. receivable at the end of every year.

I shall now show that the preceding rule agrees with
that of Mr. Milne; which is as follows: —Let P be worth
an annuity certain of ¢ years, and let v be the present
value of 1/. to be received a year hence. Then the pre-
sent value of all the fines, according to Mr. Milne, is
A’ +B’+C'+

1 avg t Th

where A’, B’, C’, &c. mean the present value of 1/., to
be received at the end of the years in which the lives
severally drop. Since P is the value of an annuity certain
for ¢ years, we have

Paboee, re Pale! opps eR,
F. v

and ee whence AS)
E+! { upthh: ak

from which the coincidence of the two rules is manifest.
The error of the old rule, by the Northampton Tables,
and at 4 per cent. (the best life being worth 17°25 years
purchase) is, that the result is 54 per cent. too great.
The old rule, as Mr. Milne justly observes, is derived
from the supposition that the new life is put in at
the beginning of the year in which the old one drops,
instead of at the end ; which last was in the intention of
those who formed the rule. It may be said however, that
¥ 2

XViii APPENDIX THE SECOND.

the old rule is nearer the truth than the new; since one
time with another, the renewal is made before the end
of the year in which the old life drops. This objection
must be valid to some extent; and I proceed to inquire
how much weight must be allowed to it.

Let a be the fraction of a year allowed for renewal :

nip l
it is clear then that — renewals (each accompanied by

a fine) may take place in the year. Let the life A drop
at the expiration of the fraction 0 of a year, or between
§ and 6+.d0, the chance of which is d@ itself, if A be
supposed equally likely to drop at any period of the year.
At 0+a, then, the new life is put in; and if this new
life drop before 1—a of the year is gone, another fine
must be paid, and another renewal is made, which again
may drop before 1—a, and so on. But since the chance
of each additional renewal is very much smaller than
that of the preceding, it will be sufficient to take the
first only into consideration. Let it be supposed, then,
that not more than one renewal shall take place within
the year in which A drops.

Let a be the chance that the life P drops in a year
after nomination, in which case we may call wa the
chance that it drops inany fraction w of a year. Then
d@(1—6—«)a is the chance that the life A drops
between 6 and 6+d 46, and that the next life drops
within the year, in which case another fine is to be paid
at the beginning of the next year. Consequently, neg-
lecting the interest of the fine in a fraction of a year,
the lessee has the chance d@ (1—@— a) a of having a
second fine to pay, upon the contingency of A dropping
between Oand 6+d6. Integrate this expression from
6=0 to 0= 1~—a, and we have 4a (1—a)? for the
chance of a second fine: which, with the fine certain
upon the death of A, shows that 1+ ta (1—a)? is the
mathematical expectation of the fines to be paid, when
the probability of one renewal within the year is con-
templated, and another at the end, if necessary.

A more complicated process, proceeding on the sup-

VALUE OF COPYHOLDS. xix

position that any given number of renewals may take
place within the year in which A dies, gives the follow-
ing terms for the total probability of one or more re-
newals taking place before the end of the year : —

a, the chance that there shall be no renewal “
1 —2a)? :
ie _ AES the chance of one only ;

a (1—2a)? a 2 (1—3a)8
rae Rie. a

and so on; the series being continued as long as the requisite
multiples of a are less than unity. Hence the chance that the
number of renewals shall not exceed n, is
an (1 —(n+1)a\"*1
SiS oc oe MSI

which, in the case most against us, that is, supposing instanta-
neous renewals, or a=0, is

the chance of two only;

a”

ie \
2eS. cee Remtl

Let the life P be one of seven years old, in which
case the Northampton table gives a=149. = +0186.
Neglecting interest for the fractions of the year, and
remembering that the number of renewals is the num-
ber of fines, the mathematical expectation of all the
fines is the sum of

eee eheeel 2 (See Bey &e.

To this add a, the chance that the renewal fine certain
on the death of A, shall outrun the year, and we find
ey 9 2 2 mae 3
ey oa 2 a) ~ #20 8a) artes
4 2 ° 3

for the mathematical expectation of all the fines which
shall be paid in the year in which the present life drops,
including the chance of that life dropping too late to
renew it within the year, and of its being therefore re-
newed within a period not exceeding @ of the year fol-
lowing. If «=O, this becomes

y 3

XX APPENDIX THE SECOND.

Slack. e being 2+718 eee
— > ga 2818
a

and when a= °0186 , e#=1°019

Hence :019~:0186 being 1°02, it appears that
1° 02(1 +7) is an enormously exaggerated representation
of the fine which must be substituted for 1/ in the amended
rule to make the correction which might be suggested
by the advocates of the old rule. Allowing r= °035,
it may be granted that the old rule is correct, in the
particular case before cited, if wesuppose: 1. That no
time whatever is allowed for renewal ; 2. That the best
life which can be found is such, that 1 out of 59 of such
lives drop ina year, and; 3. That the lessor is toreceive
his fines at the beginning of the year succeeding that in
which A drops, with 34 per cent. interest, reckoned from
the beginning of the preceding year. To take a more
rational supposition, let us allow six months for re-
newal. Here a= and only the first term of the series
can be taken, which, with the interest, gives 1+.

It appears from the preceding, that the advantage of
the lessor over that which the rule gives him, is trivial,
except in this, that at the time when the rule supposes
him to receive one pound, he may have received and
improved one pound during a fraction of a year. There
is also another advantage which he has, and which neither
rule allows him. The renewal may have been made
before the time supposed in the rule, in which case the
existing life will be somewhat worse than that supposed.
To take all these circumstances into account, suppose
the life A to drop at the end of @ of the year, in which
case the lessor is in possession of 1+r—(6+a)7 at the
end of the year, and the lessee has a life* worth

P— $1—(6+a)}aP, whereAP is the decrement

* Let it be remembered that the renewal may take place after the
beginning of the year, which is equivalent to having a better life than
that supposed. Algebra, as in other cases, strikes the balance | of
positive and negative quantities without the necessity of introducing
several formule,

VALUE OF COPYHOLDS. XXi

of the value of the life in one year. Multiplying by
d6, the chance of this occurrence, and integrating from
6=0 to @=1, we find 1+4*~—a~,r for the mean sum,
and P—4aP+a4P for the mean life. If the time
allowed for renewal be more than six months, the rule
(without this correction) gives an advantage to the Jessor ;
if six months, to neither ; if less than six months, to the
lessee. Substitute 1+4r—ar and P—4} AP+aAP
for 12 fine and a life P, and we have

E—A

; x. 1+ir—a a
1+P—jAP+aAP sunt’.

which I believe to be the most correct rule that can be
given for the present value of all the fines upon a single
life renewable for ever, the value of the present life
being A, the tabular value of the best life P, that of
a life one year older P—AP, the fraction of a year
allowed for renewal a, 7 the interest of one pound for
one year, and E the perpetuity. By substituting n E—
(A+B+C-+°:::) for E—A, the value of all the fines
from an estate held on any number of lives is found.
Similar considerations apply to the present value of 12.
to be received in a of a year after the death of a life A.
The offices frequently pay in three months after the
death is proved, whereas the tables are calculated for
the end of the year of death. Again, they rate all lives
as they will be at the next birthday, the parties being
one with another half a year younger. To a party who
dies at the end of 6 of a year, the office has paid by the
end of the year 1+ (1—a—6) r which, treated as be-
fore, gives 1+4r—ar. And the present value of 1/.,
which is computed by the office from
E—A
E+1
(where & is the proportion of profit demanded) may be
more strictly computed from
E—A-—jAA
E+ 1

(1 +2)

x (1 +4r—ar)
¥ 4

Xxll APPENDIX THE THIRD.

where A+ AA is the value of a life one year younger
than the office age of the party at entry.

When an annuity is granted upon condition that the
executors of the party are to receive such a proportion
of payment for the year in which the annuitant dies, as
corresponds to the portion of the year during which he
is alive, the addition to A the value of the annuity is
the present value of $+ 7 of a year’s purchase, pay-
able at the end of the year of death. The formula
which then very nearly represents the result of the pre-
ceding correction is

A (1 —§) +4-§
or to the tabular value of the annuity add the excess of
half a year’s purchase over half a year’s interest of the
tabular value, together with one-third of the interest of 12.

APPENDIX THE THIRD.

ON THE RULE FOR DETERMINING THE PROBABILITIES
OF SURVIVORSHIP.

Ir does not seem to have been noticed, that this rule
is considerably more correct than its framers could have
anticipated, supposing them to have contemplated no
higher degree of exactness than their demonstration en-
titled them to assert. Let dt and Wé represent the pro-
babilities that A and B, now alive, shall be alive at the
end of ¢ years, ¢ being whole or fractional. Then
—w’t.dt is the probability that B shall die between
t and t+dt and —/ot. Wt dt from t=n to t=n+1
is the chance of a survivorship of A beginning to take
place somewhere in the (n+1) th year after the present
time. Let a anda+Aa and £ and 6+A/ be the
chances that A and B shall be alive at the end of n and
a-+1 years from this time: then, in the demonstration

ee Le aes ii : os

RULE FOR SURVIVORSHIP. XXiil

of the rule, ¢¢ is assumed =a + Aa.¢ and Yt=G+A(.t,

t being measured from the beginning of the (n+1)th
year. Hence

—fot.Lt.dt (fromt=0 tot=1) = —aAB—lIAcAB

which is the common rule in a different form.
Let us now suppose (¢ being measured from the be-
ginning of the (n+1) th year)
gt=a+Aa.t + Aa. ¢"—"

Li=B+AB.t +A76.25—?

where the differences constitute series of rapidly di-
minishing terms. The only term of the second order
which this addition to the hypothesis introduces into
—f{ ot Vt.dt is aA?B f (t—4) dt which is =0, when
taxen from t=0 to t=1. Consequently the errors of
the rule are all of the third order.

To give a notion of the amount of error, extend the
preceding formule to terms of the third order, and form
the integral, reserving only the terms of the third order.
The final result is as follows :—If x and y be the num-
ber of the living at the age of A and B, andif a, 6, ¢,...
be the numbers alive, atn,n+1,... years older than A,
and p, q, r,...atn,n-+1, ... years older than B, then
the probability that A shall begin to survive B in the
course of the (m+ 1) th year of the calculation, is

(2+b)(p—q) . (b—c)(p—q) — (a—b)(q—r)
2Qary 12zy

the first term being that generally used, and the second
a correction which ought always to be applied in those
parts of the table in which the yearly decrements are
not equal.

The demonstration of the preceding will be easily
arrived at by the indication which I have given, by any
one acquainted with the integral calculus. To those
who have not that advantage, reason may be shown in
the result, though not for the result. The preceding

XXiv APPENDIX THE FOURTA.

correction will have the positive sign when b—c bears a
greater proportion to a—h than doesq—rto p—q:
that is, when the mortality in A’s part of the table is in-
creasing faster than in B’s. Now, ceteris paribus, the
larger the comparative mortality of the year succeeding
a given -year, the more likely are the deaths of the latter
part of that given year to predominate over those of the
former ; consequently, the more likely is the death of
A, if it happen in that year, to be towards the end of it.
But any thing which shows that the death of A is more
likely than before to take place later in its year, increases
the probability that a survivorship commencing in that
_ year shall be in favour of A, and not of B.

APPENDIX THE FOURTH.

ON THE AVERAGE RESULT OF A NUMBER OF OB-
SERVATIONS.

Tuat I might not further embarrass the most ab-
struse chapter of this work, by the introduction of an
isolated point of difficulty, I have chosen here to men.
tion some considerations connected with the value of
the average of observations. There is a remarkable
difference of principle between two problems which at
first sight appear identical; namely, where it is re-
quired to invent a method of treating observations be-
fore they are made, and after they are made. Positive
and negative errors being equally likely, and no observ-
ations having been made, it is easily proved that there
is a high probability in favour of a large number of
observations giving exactly or nearly the same total
amount of one as of the other, The case is analogous
to that of an urn filled with black and white balls in

ON THE AVERAGE RESULT OF OBSERVATIONS, XX¥V

equal numbers, out of which, in a large number of
drawings, both sorts will come in nearly equal propor-
tions

But in this, as in every other question of proba-
bilities, any additional knowledge of the circumstances
which may happen, or have happened, changes the
problem, and is equivalent to an extension or limitation
of its conditions, When the observations have been
made, the position of the observer is altered, since though
the law of facility of error be not determined, yet more
probability is given to some laws than to others, by
inspection of the observations themselves. For instance;
if the observations give results of very little discordance,
it is immediately obvious that a law of facility which
makes the probability of large errors very small, is
more likely to have been that which actually existed
than one of a different character. The problem now
presents an analogy with that of an urn, from which
drawings have been made and registered, so that the
contents of the urn are to be guessed at from the
drawings.

In the first problem, and supposing that a method of
combining the observations is to be chosen before obser=
vations made, it is demonstrable that the average of the
results is more likely to be true than any other magni-
tude. And the same conclusion seems probable in the
second case, since unassisted common sense would never
draw any distinction between the two problems, But the
results of calculation applied to the development of the
distinction just drawn, show that the average of obser-
vations made is not necessarily the most probable re-
sult, nor can be such for more than two observations,
unless one particular law of facility of error be sup-
posed, which law is the standard law described in
Chapter VII. But it is also shown, as mentioned in
page 142, that the results of any law of facility, when
applied to tolerably large numbers of observations, are
nearly identical with those of some variety or other of
the standard law; so that, practically, the average of

XXV1 APPENDIX THE FIFTH-

observations is either the result which the strictest ap-
plication of sound principles would declare to be the
most probable truth, or else very near to it. !

It is,in themeanwhile, a most remarkable circumstance
that a method so simple, and so conformable to common
sense, as that of averaging, should first turn out to be
incorrect, except upon a supposition never contemplated
in thinking of the evidence of this rule, and should after-
wards prove to be always nearly correct, for large num-
bers of observations, on account of the tendency of all
admissible suppositions to confound themselves, as the
number of observations increases, with that one parti-
cular supposition, which makes the common notion
absolutely correct. My own impression, derived from
this and many other circumstances connected with the
analysis of probabilities, is, that mathematical results
have outrun their interpretation: and that some simple
explanation of the force and meaning of the celebrated
integral, whose values are tabulated at the end of this
work, will one day be found to connect the higher and
lower parts of the subject with a degree of simplicity
which will at once render useless (except to the his-
torian) all the works hitherto written.

APPENDIX THE FIFTH.

ON THE METHOD OF CALCULATING UNIFORMLY DE«#
CREASING OR INCREASING ANNUITIES.

An authority from which I rarely differ has spoken
thus, “‘ A few writers on these subjects, of late years,
have employed the differential and integral calculus in
their investigations. We have not yet seen any fruits

ON UNIFORMLY CHANGING ANNUITIES. ° XXVii

of this application of the calculus, which appear to us
of much value, nor are we at all sanguine in expecting
any.” ‘he tendency of such an assertion is to en-
courage those who study the subject, to stop short of
the differential calculus _in their mathematicai studies.
Now JI assert, 1. That the calculus aforesaid may, as
evidenced in the results of chapter IV., lead to most
valuable rules in the estimation of complicated proba-
bilities. 2. That if the calculus be not serviceable in
the deduction of the law of mortality, it is from defect
of observed data. As soon as larger and more correct
tables of the numbers living are obtained, the differen-
tial calculus is ready to furnish methods for correcting
those now in use. 3. That the differential calculus may
be made to give important simplifications of processes,
and to render the tables already constructed immediately
available for purposes to which no one now dreams of
applying them.

If v be the present value of 1/., to be received at the
end of a year, and gv be the present value of a con-
tingent annuity of 1/., then that of an annuity which is
to be 1/. at the end of the first year, 2/., 31., &c., at
the end of the second, third, &c. years, is vp'v, where g'v
is the differential coefficient of gv. Now 1+7 being
the amount of 1/. in one year, we have

d 1 d d
scan =-lll See ; ve = re (1 +r)2
dr (1 +r)? dv dr
; dov dov
t — = — —
and the annuity v— = (1 +r)

Now tables of annuities of 1/. being calculated for a
succession of values of 7 differing by ‘01, we have -

dou
Ol x — Agv — 1 A%pv + } A3gv — 1 Atov + Ke.
T

ad ;
Substitute the value of thence obtained, and we

XXVili APPENDIX THE FIFTH.
have a method of finding the value of the required
annuity, which may be described in the following

RULE.

Take out the value of an annuity of 1/7. at the given
rate of interest, and at several successive higher rates:
take the successive differences, the difference of the dif-
ferences, and so on. To the first difference add half of
the second difference, one-third of the third, and so on:
the sum of these, multiplied by the amount of 100/. in
one year at the first named rate, is the value of the
annuity required.

I take examples from the Northampton tables, at 4
per cent., because Mr. Morgan has given a table of the
annuities required, which will serve to find verifications.
First suppose the age to be 5 years.

Annuity 4p.c. 17:248 9°49]

— 5 — 14827 Jigen 556 64

—— 6 — 12:962 j.47g "392 "394 060
— 7— 11489 jug, 288

ce 8s — 10°304

2°4921 + lof 556 + 4 of 164 + 4 of 060 = 2°769
2°769 x 104 = 288°'0 answer: in Morgan 288°4

Next suppose the age to be 80 years,
Annuity 4p.c. 3°643

fo 9615: oe On
— 6 — 3394 74 ‘008
— 7 — 3281 5° ‘006
etnehica/ Bs em” SALTS

128 + 3 0f 007 = °132; °132 x 104 = 13-7 answer
13°8 in Morgan.

Prosiem. A life annuity is £m at the end of the
first year, and diminishes £n every year, until nothing
is due, after which it ceases entirely. Required its
present value.

Rute. When n is so small, that the annuity cannot

ON UNIFORMLY CHANGING ANNUITIES. Xxix

be extinguished during the tabular life of the party,
from the value of an annuity of £(m-+n) subtract n
times that of an increasing annuity of 1/. found as
already described. But when the annuity can be ex-
tinguished during the life of the party (say in ¢ years
exactly, so that m=nt), then to the preceding result
add n times the value of an increasing annuity of 1/.
on a life ¢+1 years older than the party, multiplied by
the chance of his living ¢+1 years, and by the presen:
value of £1 due ¢+1 years hence.

Prositem. Required the present value of £m to be
received at the end of the year in which A dies, if in
a year, or £(m—n) if in the second year, and so on,

Rutz. When 7 is so small that the sum insured
cannot be extinguished during the tabular life of the
party, to the value of a perpetuity of £m, add that of
an increasing annuity of £n, £2n, £3n, &c., and
subtract the value of a simple life annuity, of which
the yearly payment is £m, increased by the product of
a perpetuity due, and the value of a simple annuity of
#£n: divide the difference by the value of a perpetuity
due, and the quotient is the present value required. But
if the insurance be extinguished in ¢ years, or if m=nt:
find the product of an annuity due of £1 ona life +1
years older than the given life, and of a perpetuity ;
subtract the value of an increasing annuity of 1/. on that
life, and having multiplied the difference by the chance
of the first life surviving ¢+ 1 years, and by the present
value of £n due ¢+1 years hence, add the result to the
dividend in the first part of the rule, before dividing by
the value of a perpetuity due of £1.

Various other questions will present themselves, which
can be easily reduced to practice by aid of the expedi-
tious rule for finding the values of increasing annuities.
This rule may be applied to the Carlisle tables (for
which Mr. Milne has deduced the values of annuities
on single lives, at rates of interest from 3 to 8 per cent.,

XXX APPENDIX THE FIFTH.

both included), and also to joint lives, though not —
’ with so much correctness, on account of the tables not —
containing so many rates of interest.
The following is an instance in which a deduction from
the calculus of differences will supply in a rough man-
ner the deficiencies of tables. There are none of these
for determining the mean duration of the joint existence
of two lives, but the defect may be supplied with suf-
ficient accuracy for many purposes, and particularly at
the middle and older ages, by tlre following Rue.
Let (3), (4), &c. stand for the values of an annuity
on a single life, or on two joint lives, at 3, 4, &c.
per cent. : from twice (3) subtract (6) and reserve the
remainder: from (4) subtract (5), and having halved
the remainder, to it add the tenth part of (5), and
multiply the result by 9. Subtract the last product
from the reserved remainder, and multiply the differ-
ence by 10. The result increased by °5 in the case of
a single life, or by °25 in that of two joint lives, will
be something under the mean duration required. For
example, and to take a very unfavourable case, let the
Carlisle table be used, the life being 10 years old.

(8) = 23-512 (4) = 19°585
2 (5) = 16°669
47-024 2) 2:916
(6) = 14-448 ——
1°458
32°576 i (5) 1-667
28°125
3125
4°451 9
10
28°125
44°51
5

45°01 the truth being 48°82

Sn Page tet ya

ee Oe ge ae, ee
- eee

ON OFFICE VALUATIONS. XXXi

The error diminishes as the age increases, as the
following table will show : —

Age.|Approx.| Truth. | Age. |Approx.} Truth,

O | 34°78 | 38°72 § 60 | 14°13 | 14:34
10 | 45°01 | 48°82 § 70 9°33} 9°18
20 | 39°07 | 41°46 §-80 | 5°57) 5°51
30 | 38°09 | 34°34 7.90 | 38°43] 3°28 >
40 | 27°05 | 27°61 #100 | 2°36] 2°28
50 | 20°96 | 21°11

APPENDIX THE SIXTH.

ON A QUESTION CONNECTED WITH THE VALUATION OF
THE ASSETS OF AN INSURANCE OFFICE.

Iv an insurance office were about to close its doors,
and to buy up all the policies of its members, the pro-
cess of valuation would only require the assets to be ex..
pressed by the amount of money which they would
actually produce at the time of valuation. In this
case, the profit or surplus is properly expressed by
A+P—C, or the amount of assets increased by the
present value of all the premiums, and diminished by
that of all the claims.

But valuation is not usually made with reference to
an immediate settlement ; but for the purpose of ascer-
taining what sum can be set apart as profit, and de-
clared to belong to existing policies, without anticipatory
injustice to future members. The preceding formula,
with allowance for expences of management, still repre-
sents the sum which may be called profit, provided that
the stock belonging to the office can really be improved at
the rate of interest assumed in the valuation. For the

sufficiency of this stock to answer all demands depends
_ upon its increasing at that rate of interest upon which

the values of P and C were found.
Z

XXXil APPENDIX THE SIXTH.

Now, it generally happens, that the property of an
insurance office consists of funds invested at different
rates of interest, the consequence of which is, that there
is no absolutely rigorous method of determining the
profit, except by prospective calculation of the state of
the office for every year of the tabular duration of the
life of its youngest member. Supposing the insured to
die precisely in the manner indicated in the table, and
assigning the order in which the different principals are
to be touched, when necessary, it is then possible to
calculate the amount which will remain when all claims
are paid. The present value of this amount (the spe-
cies of stock in which it is to be left being known) is
all that can be called profit at the time of the valuation.
This process, however, is exceedingly laborious ; and,
in all probability, where yearly valuations are made, the
expence of making the calculation would be greater than
the loss prevented by taking the more simple, but less
accurate, method.

If money made only simple interest, and computations
were performed accordingly, no difficulty would arise:
for £S improving at » per pound, and £9’ at 7’
per pound, is at all times equivalent to #(S+8’), im-
proving at (Sr+S’ 7”) + (S+8’) per pound: so that
all the different stocks might be considered as lying at
one average rate of interest. Such, however, is not the
case with compound interest.

To introduce the question in a simple form, let us sup-
pose that all the stock of the office makes r per pound,
the rate assumed in the valuation, except only one sum,
H, which makes r’ (less than 7) per pound. If, then,
this sum were set down as H in the item A, the profit
would be overrated ; nor can we answer the question,
how much should it be estimated at, without some re-
ference to the time at which H, with its accumulations,
is to become necessary. If this will not be wanted for
n years, then H x (1+7’)"+(1+7r)" is the value at
which it must be estimated.

ON OFFICE VALUATIONS. XXXili

The best method of treating this case is to suppose
H to stand for such a sum, that there will be no loss
arising from a lower rate of interest before the next
valuation. Accordingly, in the preceding formula,
must be the number of years intervening between two
valuations. If such a process should give too little
profit at one valuation, the same item will be larger in
the next, and vice versd: so that there will be a con-
tinual tendency to correctness. If, for instance, the
valuations be made yearly (for which this very cir-
cumstance is one reason among many), then H (1+7’)
—(1+7) should be taken for H, and the existing poli-
cies may have the benefit when 1’ is greater than 7.

XXXIV
TABLE I.
H. A A? t. H A
0°00 | 0:00000 00 | 1128 33 22 | 0°40 | 0°42839 22 | 957 68
0°01 | 0°01128 33 | 1128 11 45 | 0-41 | 0°43796 90 | 949 86
0°02 | 0°02256 44 | 1127 66 67 | 0°42 | 0:44746 76 | 941 91
0°03 | 0°03384 10 | 1126 99 90 | 0°43 | 0°45688 67 | 933 84
0°04 | 0°04511 09 | 1126 09 | 1 12 § 0°44 | 0°46622 51 | 925 67
0°05 | 0°05637 18 | 1124 97 | 1 35 f 0°45 | 0°47548 18 | 917 37
0°06 | 0°06762 15 | 1123 62 | 1 58 § 0°46 | 0°48465 55 | 908 97
0:07 | 0°07885 77 | 1122 04 | 1 79 | 0°47 | 0°49374 52 | 900 46
0°08 | 0°09007 81 | 1120 25 | 2 O1 | 0°48 | 0°50274 98 | 891 85
0°09 | 0°10128 06 | 1118 24 | 2 24 # 0°49 | 0°51166 83 | 883 16
0°10 | 0°11246 30 | 1116 00 | 2 46 f 0°50 | 0°52049 99 | 874 38
O-1l | 0°12362 30 | 1113 54 | 2 67 } 0°51 | 0°52924 37 | 865 50
0°12 | 0°13475 84 | 1110 87 | 2 88 § 0°52 | 0°53789 87 | 856 54
0°13 | 0°14586 71 | 1107 99 | 3 10 f 0°53 | 0°54646 41 | 847 51
0°14 | 0°15694 70 | 1104 89 | 3 31 § 0°54 | 0°55493 92 | 838 41
0°15 | 0°16799 59 | 1101 58 | 3 52 | 0°55 | 0 56332 33 | 829 24
0°16 | 0°17901 17 | 1098 06 | 3 72 § 0°56 | 0°57161 57 | 820 01
0°17 | 0°18999 23 | 1094 34 | 3 93 | 0°57 | 0°57981 58 | 810 71
0°18 | 0°20093 57 | 1090 41 | 4 14 § 0°58 | 0°58792 29 | 801 36
0°19 | 0°21183 98 | 1086 27 | 4 34 # 0°59 | 0°59593 65 | 791 96
0°20 | 0°22270 25 | 1081 93 | 4 53 | 0°60 | 0°60385 61 | 782 51
0°21 | 0°23352 18 | 1077 40 | 4 73 ¢ 061 | 0°61168 12 | 773 02
0°22 | 0°24429 58 | 1072 67 | 4 92 | 062 | 0°61941 14 | 763 49
0°23 | 0°25502 25 | 1067 75 | 5 12 § 0°63 | 0°62704 63 | 753 94
0°24 | 0°26570 00 | 1062 63 | 5 29 § 0°64 | 0°63458 57 | 744 35
0°25 | 0°27632 63 | 1057 34 | 5 49 | 0°65 | 0°64202 92 | 734 73
0°26 | 0°28689 97 | 1051 85 | 5 67 § 0°66 | 0°64937 65 | 725 10
0°27 | 0°29741 82 | 1046 18 | 5 84 § 0°67 | 0°65662 75 | 715 45
0°28 | 0°30788 00 | 1040 34 | 6 O1 | 0°68 | 0°66378 20 | 705 79
0°29 | 0°31828 34 | 1034 33 | 6 19 | 0°69 | 067083 99 | 696 11
0°30 | 0°32862 67 | 1028 14 | 6 36 § 0°70 | 0°67780 10 | 686 44
0°31 | 0°33890 81 | 1021 78 | 6 52 § 0°71 | 0°68466 54 | 676 76
0°32 | 0°34912 59 | 1015 26 | 6 67 | 0°72 | 0°69143 30 | 667 08
0°33 | 0°35927 85 | 1008 59 | 6 84 § 0°73 | 0°69810 38 | 657 42.
0°34 | 0°36936 44 | 1001 75 | 6 98 | 0°74 | 0°70467 80 | 647 76
0°35 | 0°37938 19 | 994 77 | 7 14 § 0°75 | 0°71115 56 | 638 11
0°36 | 0°38932 96 | 987 63 | 7 29 | 0°76 | 0°71753 67 | 628 49
0°37 | 0°39920 59 | 980 34 | 7 42 | 0°77 | 0°72382 16 | 618 &8
| 0°28 | 0°40900 93 | 972 92 | 7 55 | 0-78 | 0°73001 04 | 609 31
0°39 | 0°41873 85 | 965 37 | 7 69° O79 | 0°73610 35 | 599 75

pay doi pin delalndag Ans diese ceed,

Pee sae

ee eae

TABLE I. XXXV
| ' is
t. | aed eee | A? | t. H. ager: A? |
1 i {
if ‘ t -_—

0:80 | 0°74210 10 , 590 23 | 9 48 | 1-30 | 093400 80 | 205 52 | 5 a2
0°81 | 0-74800 33 | 58075 9 45] 1-31 | 093606 32 200 20 | 5 22
0°82 | 0°75381 08 571 80 9 41 | 1°32 0-93806 52) 194 98 | 5 11
0°83 | 0:75952 38 , 561 89 9 36] 1:33) 094001 50! 189 87 , 5 02
0-84 | 0°76514 27 552 53 | 9 31 | 1°34 | 0°94191 37 | 184 85 | 4 93

| 1
0°85 | 0°77066 80 | 543.22 | 9 26 | 1°35 , 0:94376 22 179 92 | 4 82
0°86 | 0°77610 02 | 533 96 | 9 21 | 1:36 | 0°94556 14. 175 10 | 4 74
0°87 | 0°78143 98 | 524 75 | 9 16 | 1°37 | 0-94731 24 | 170 36 | 4 63
0:88 | 0°78668 73 | 515 59 | 9 09 | 1°38 | 094901 60; 165 73 | 4 55
0°89 | 0°79184 32} 506 50 | 9 04 | 1:39 | 0-95067 33 | 161 18 | 4 45
090 | 0°79690 82 | 497 46 | 8 97 | 1:40 | 0°95228 51 | 156 73 | 4 35
0-91 | 0*80188 28 | 488 49 | 8 91 | 1:41 | 0°95385 24 | 152 38 | 4 27
0-92 | 0°80676 77 | 479 58 | 8 83 | 1:42 | 0:95537 62 | 148 11 | 4 18
0-93 | 0°81156 35 | 470 75 | 8 77 | 1°43 | 0-95685 73 | 143 93 | 4 09
0°94 | 0°81627 10 | 461 98 | 8 70 | 1:44 | 0:95829 66 | 139 84 | 3 99
0:95 | 0°82089 08 | 453 28 | 8 61 | 1-45 | 0:95969 50 | 135 85 | 3 91
0:96 | 0°82542 36 | 444 67 | 8 55] 1-46 | 0:96105 35 | 131 94 | 3 82
0:97 , 0°82987 03 | 436 12 | 8 46 | 1-47 | 0:96237 29 | 128 12 | 3 74
0-98 | 0°83423 15 | 427 66 | 8 39 | 1°48 | 0-96365 41 | 124 38 | 3 65
0:99 | 0°83850 81 | 419 2Z | 8 30 § 1-49 | 096489 79 | 120 73 | 3 57
1:00 | 0-84270 08 | 410 97 | 8 22] 1°50 | 0°96610 52 | 117 16! 3 49
1-01 | 0-84681 05 | 402 75 | 8 13] 1-51 | 0:96727 68 ; 113 67 | 3 40
1:02 | 0-85083 80 | 394 62 | 8 05 | 1°52 | 0:96841 35 | 110 27 | 3 32
1:03 | 085478 42 | 386 57 | 7 96 | 1°53 | 0-96951 62 | 106 95 | 3 25
1°04 | 0°85864 99 | 378 61 | 7 86 | 1:54 | 0-97058 57 | 103 70 | 3 16
1:05 | 0:86243 60 | 370 75 | 7 774 1:55 | 0°97162 27 | 100 54 | 3 09
1:06 | 0°86614 35 | 362 97 | 7 68 | 1-56 | 0:97262 81} 97 45 | 3 Ol
1:07 | 0°86977 32 | 355 29 | 7 60 | 1-57 | 0:97360 26 | 94 44 | 2 94
1:08 | 0°87332 61 | 347 69 | 7 49 } 1:58 | 0:97454 70} 91 50 | 2 86
1:09 | 0°87680 30 | 340 20 | 7 40 | 1-59 | 0°97546 20] 88 64 | 2 79
1:10 | 0:88020 50 | 332 80 | 7 31 | 1:60 | 0:97634 84 | 85 85 | 2 73
1:11 | 0°88353 30 | 325 49 | 7 21 | 1-61 | 0:97720 69} 83 12 | 2 64
1:12 | 0°88678 79 | 318 28 | 7 12 | 1-62} 0-97803 81} 80 48 | 2 59
1:13 | 0°88997 07 | 311 16 | 7 O01 | 1:63 | 0:97884 29} 77 89 | 2 51
1:14 | 0°89308 23 | 304 15 |-6 91 | 1°64 | 0°97962 18] 75 38 | 2 45
1:15 | 0°89612 38 | 297 24 | 6 82 | 1:65 | 098037 56 | 7293 | 2 38
1:16.| 0°89909 62 | 290 42 | 6 72} 1:66 | 0-98110 49 | 70 55 | 2 31
1:17 | 0-90200 04 | 283 70 | 6 61 § 1°67 | 0-98181 04 | 68 24 | 2 26
1°18 | 0°90483 74 | 277 09 | 6 &2} 1°68 | 0:98249 28} 65 98 | 2 20
1:19 | 090760 83 | 270 57 | 6 42} 1-69 | 0-98315 26 | 63 78 | 2 12
1:20 | 0°91031 40 | 264 15 | 6 31 | 1°70 | 0°98379 04 | 61 66 | 2 08
1:21 | 0°91295 55 | 257 84 | 6 22] 1-71 | 0°98440 70 | 59 58 | 2 O1
1:22 | 091553 39 | 251 62 | 6 11 | 1:72} 098500 28} 57 57 | 1 96
1:23 | 091805 O1 | 245 51 | 6 02} 1-73 | 098557 85 | 55 61 | 1 90
1:24 | 0°92050 52 | 239 49 | 5 91 1-74 | 098613 46} 53 71 | 1 85
1:25 | 0°92290 01 | 233 58 | 5 81 | 1:75 | 0-98667 17 | 51 86 | 1 79
1:26 | 0°92523 59 | 227 77 | 5 71 | 1°76 | 0:93719 03 | 50 07 | 1 75
1-27 | 0-92751 36 | 222 06 | 5 GL} 1-77 | 0:98769 10] 48 32 | 1 68
1:28 | 0:92973 42 | 216 45 | 5 52] 1-78 | 0-98817 42] 46 64 | 1 65
1-24 | 0°93189 87 | 210 93 | 5 41 | 1:79 | 0-98864 06} 44 99 | 1 59

N
09

TABLE I.

RS
7

H.

One

—— i
SSR FSSLS

=—S CONDO PWNKO COD

RIDIN RII WNWNWNN wWrnMMn~p MMM wrorwiyry
SI9KNNY NWNWWN bp by bP BD bP

nD bo ee

oe

bo
_

don wh we
OOBNO

0°98909 05
0°98952 45
0°98994 31
0°99034 67
099073 59

0°99111 10
0°99147 25
0°99182 07
0°99215 62
0°99247 93

0°99279 04
0°99308 99
0°99337 82

0°99365 57 :

0°99392 26

0°99417 94
0°99442 63
0°99466 37
0°99489 20
0°99511 14

0°99532 23
0°99552 48
0°99571 95

0°99590 63 |
0:99608 58 ,

0°99625 81 ;
0:99642 35

0°99658 22
0°99673 44
0°99688 05

0°99702 05

0°99715 48 .

0°99728 36
0°99740 7C

0°99752 53 |
0:99763 86

0°99774 72
0°99785 11
0°99795 07
0°99804 59

0°99813 72
0°99822 44
0°99830 80
0°99838 78
0°99846 42

0°99853 73
0°99860 71

0°99867 39
0°99873 77 °
0°99879 86 ©

ee

aa

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iL aenliaeiLansiLand
—
rs

1 7k WO Wo oe)

foment fees peed poet Demet ed feed pet et
_
fer)

WATT AAADH AQHGAOSD aagaan aghann PARRA ARAKRER GHHBMWHD Www
RONDO DONST BONWHKS BDHYSHR BEWHS SONAHG PONHS GSGHNAG PWONHKS

f=?)
—
IWNNNIN WHNNNY NWNNWHNY WHWNNWID WNWNWNHNY wWiytrIp WNNNNY wIdwI NHL

yaa yy
CO OWIA HN

099885 68
0°99891 24
0°99896 55
0°99901 62
0:99906 46

0°99911 07
099915 48
0°99919 68
0°99923 69
099927 51

0°99931 15
0°99934 62
0°99937 93
0°99941 08
0°99944 08

0°99946 94
0°99949 66
0°99952 26
0°99954 72
0°99957 07

0°99959 31
0°9996i 43
0°99963 45
0°99965 37
0°99967 20

0°99968 93
0°99970 58
0°99972 15
0°99973 65
0°99975 06

0°99976 40
0°99977 67
099978 88
0°99980 03
0°99981 12

0°99982 15
0°99983 13
0°99984 06
0°99984 94
0°99985 78

0°99986 57
0°99987 32
0°99988 03
0°99988 70
0°99989 33

0°99989 94
0°99990 51
0°99991 05
0°99991 56
0°99992 04

mt et et
anoeaoe

Oneww AYOLAL KHPTABATT AHMAD wWTo1380

TABLE 1. XXXVli

t. H. A A? t. A. A A?
2°80 | 0°99992 50 43 29 2°91 | 0°99996 13 23 1
2°81 | 0°99992 93 4] 3 | 2°92 | 0°99996 36 22 1
2°82 | 0°99993 34 38 1 | 2°93 | 0°99996 58 21 2
2°83 | 0:99993 72 37 3 f 2°94 | 0°99996 79 19 1
2°84 | 0°99994 09 34 1

2°95 | 0°99996 98 18 1
2°85 | 0°99994 43 33 2 | 2°96 | 0°99997 16 17 0
2°86 | 0°99994 76 31 2 § 2:97 | 0:99997 33 17 2
_2°87 | 0°99995 07 29 2 # 2°98 | 0°99997 50 15 1
2°88 | 0°99995 36 27 1 | 2°99 | 0°99997 55 14 -
2°89 | 0°99995 63 26 2

3°00 | 0°99997 79 ~ -
2°90 | 0°99995 89 24 I

*,* This table has been continued to 3°00 from the data in Kramp’s
Treatise on Astronomical Refractions. The description of it in the work
was written before this addition was made,

XXXViii

TABLE II.
j
K Att | at Avast K
0:00 0:00000 538 0°45 | 0°2385 513 0°90 0°45618
0°01 0:00538 538 0°46 | 0°24364 512 0°91 0°46064
0:02 ; 0:°01076 538 0°47 0°24876 512 0°92 0°46509
0°03 0°01614 538 0°48 0°25388 510 0°93 0°46952
0°04 | 0°02152 538 0°49 | 0°25898 509 0°94 | 0°47393
005 | 0:02690 538 0°50 | 0°26407 508 0°95 | 0°47832
0:06 | 0°03228 538 0°51 0°26915 506 0°96 | 0°48270
0°07 0:03766 537 0°52 | 0°27421 506 0°97 0°48705
0°08 | 0:°04303 537 0°53 0°27927 504 0°98 0°49139
0:09 | 0°04840 538 0°54 | 0°28431 503 0°99 : 0°49570
0°10 | 0°05378 536 0°55 | 0°28934 502 1:00 | 0°50000
O11 0:05914 537 0°56 | 0°29436 500 1:01 0°50428
0°12 | 0°06451 536 0°57 | 0°29936 499 1°02 | 0°50853
0°13 0°06987 536 0°58 0°30435 498 1°03 0°51277
0:14 | 0:07523 536 0°59 | 0°30933 497 1:04 | 0°51699
0°15 | 0°08059 535 0°60 | 0°31430 495 1°05 0°52119
0°16 | 0°08594 535 0°61 0°31925 494 1:06 0°52537
0°17 0:09129 534 0°62 | 0°32419 492 1°07 0°52952
0°18 | 0°09663 534 0°63 | 0°32911 491 1:08 0°53366
0:19 | 0°10197 534 0°64 | 0°33402 490 1:09 | 0°53778
0°20 | 0:10731 533 0°65 | 0°33892 488 1°10 | 0°54188
0°21 0°11264 532 0°66 0°34380 486 1°31 0°545S5
0°22 | 0°11796 532 0°67 | 0°34866 486 1°12 | 0°55001
0°23 0°12328 532 0°68 0°35352 483 te 5 § 0°55404
0:24 | 0°12860 531 0°69 | 0°35835 482 1:14 | 0°55806
0°25 | 0°13391 530 0°70 | 0°36317 48] 1:15 | 0°56205
0°26 | 0°13921 530 0°71 0°36798 479 1°16 | 0°56602
0:27 0°14451 529 0°72 |: 0:37277 478 ES i f 0°46998
0°28 0:14980 528 0°73 0°37755 476 1°18 0°57391
0°29 | 0°15508 527 0°74 | 0°38231 474 1:19 | 0°57782
0°30 | 0°16035 527 0°75 | 0°38705 473 1°20 | 0°58171
0°31 0°16562 526 0°76 0°39178 Ah AsSh 0°58558
0°32 | 0:°17088 526 0°77 | 0°39649 469 1°22 | 0:°58942
0°33 0°17614 524 0:78 0°40118 468 @ 1°23 0°59325
0°34 | 0°18138 524 0°79 | 0°40586 466 1:24 | 0°59705
0°35 | 0°18662 523 0°80 | 0°41052 465 1°25 | 0°60083
0°36 | 0°19185 522 0°81 0°41517 462 1°26 | 0°60460
0°37 | 0°19707 522 0°82 | 0°41979 461 1:27 | 0°60833
0°38 0°20229 520 0°83 0:42440 459 1°28 0°61205
0°39 | 0:20749 519 0°84 | 0°42899 458 1:29 | 061575
0°40 | 0°21268 519 0°85 | 0°43357 456 1°30 | 0°61942
0°41 0°21787 517 0°86 | 0°43813 454 1°31 0°62308
0°42 0°22304 517 0°87 0°44267 452 1:32 | 0°62671
0°43 0°22821 515 0°88 0°44719 A450 1°33 | 0°63032
0°44 | 0°23336 515 0°89 | 0-45169 449 1°34 } 0°68391

TABLE II. XXX1X

K A t. K A t. K A
0°63747 | 355 § 1°85 | 0°78790 | 246 | 2°35 | 0°88705 | 152
0°64102 | 352 — 1°86 { 0°79036 | 244 — 2:36 | 0°88857 | 151
0°64454 | 350 § 1°87 | 0°79280 | 242 § 2°37 | 0-89008 | 149
0°64804 | 348 § 1°88 | 0°79522 | 239 — 2°38 | 0°S9157 | 147
0°65152 | 346 § 1°89 | 0°7976 238 § 2°39 | 0°89304 | 146
0°65498 | 343 § 1:90 | 0°79999 | 236 § 2°40 | 0°89450 | 145
0°65841 | 341 1°91 | 9°80235 | 234 § 2°41 | 0°89595 | 148
0°66182 | 339 — 1°92 | 0°80469 | 231 § 2°42 | 0°89738 | 141
0°66521 | 337 § 1°93 | 0°80700 | 230 § 2°43 | 0-89879 | 140
0°66858 | 335 § 1°94 | 080930 | 228 § 2-44 0°90019 | 138
0°67193 | 333 9 1°95 | 0°81158 | 225 | 2°45 0°90157 | 136
0°67526 | 330 7 1°96 | 0°81383 | 224 — 2°46 | 0°90293 | 135
0°67856 | 328 | 1°97 | O°81607 | 221 § 2°47 | 0°90428 | 134
0°68184 | 326 — 1°98 | 0°81828 | 220 § 2°48} 0:90562 | 132
0°68510 | 323 § 1°99 | 0°82048 | 218 4 2°49} 0-90694 | 131
0°68833 | 322 | 2°00 | 0°82266 | 215 | 2°50) 0°90825 | 129
0°69155 | 319 | 2°01 | 0°82481 | 214 § 2°51 | 0°90954 | 128
0°69474 | 317 § 2°02] 0°82695 | 212 f 2°52 | 0-91082 | 126
0°69791 | 315 § 2°03 | 0°82907 | 210 § 2°53 | 0°91208 | 124
0°70106 | 313 § 2°04 | O°83117 | 207 | 2°54 | 0°91332 | 124
0°70419 | 310 § 2°05 | 0°83324 | 206 § 2°55 0°91456 | 122
070729 | 309 § 2°06 | 0°83530 | 204 § 2°56 | 0°91578 | 120
0°71038 | 306 § 2°07 | 0°83734 | 202 | 2°57 | 0-91698 | 119
0°71344 | 304 — 2°08 | 0°83936 | 201 — 2°58 | 0°91817 | 118
0°71648 | 301 § 2°09 | 0°84137 | 198 f 2°59 | 0°91935 | 116
0:71949 | 300 § 2°10 | 0°84335 | 196 § 2°60} 0°92051 | 115
0°72249 | 297 § 2°11 | 0°84531 | 195 § 2°61 | 0°92166 | 114
0°72546 | 295 § 2°12 | 0°84726 | 193 § 2°62 | 0°92280 | 112
0°72841 | 293 § 2°13 | 0°84919 | 190 | 2°63; 0-92392 | 111
0°73134 | 291 § 2°14] O°85109 | 189 4 2°64} 0°92503 | 110
0°73425 | 289 § 2°15 | 0°85298 | 188 — 2°65] 0°92613 | 108
0°73714 | 286 } 2°16 | 0°85486 | 185 § 2°66 | 0°92721 | 107
0°74000 | 285 | 2°17 | O°85671 | 183 § 2°67] 0°92828 | 106
0°74285 | 282 9 2°18 | 0°85854 | 182 # 2°68 | 0-92934 | 104
0°74567 | 280 | 2°19 | 0°86036 | 180 f# 2°69 | 0°93038 | 103
0°74847 | 277 § 2°20 | 0°86216 | 178 | 2°70 | 0°93141 | 102
0°75124 | 276 ff 2°21 | 0°86394 | 176 { 2°71 | 093243 | 101
0°75400 | 274 § 2°22 | 086570 | 175 § 2°72 | 0°92344 99
0°75674 | 271 § 2°23 | 0°86745 | 172 § 2°73 | 0°93443 98
0°75945 | 269 9 2°24! 086917 | 171 § 2°74 | 0°93541 97
0°76214 | 267 f 2:25 0°87088 | 170 § 2°75 | 0°93638 96
0°76481 | 265 | 2°26 | 0-87258 | 167 9 2°76 | 0°93734 94
0°76746 | 263 § 2°27 | 0°87425 | 166 § 2°77 | 0°93828 94
0°77009 | 261 } 2°28 | 0°87591 | 164 } 2°78 | 0°93922 92
0°77270 | 258 § 2°29 | 0°87755 | 163 § 2°79 | 094014 91
0°77528 | 257 § 2°30 | 0°87918 | 160 § 2°80 | 0°94105 90
0°77785 | 254 — 2°31 | 0°88078 | 159 f— 2°81 | 0°94195 89
0°78039 | 252 § 2°32 | 0°88237 | 158 # 2°82] 0°94284 87
0°78291 | 251 § 2°33 | 088395 | 155 § 2°83 | 0°94371 87
0°78542 | 248 # 2:34 O'88550 | 155 | 2°84 | 0°94458 85

AA

TABLE II.

SpoTTiswoopEs and S#aw,

New-street-Square.

K t. K A K
2°85 | 0°94543 3°10 | 0°96346 60 § 3°35 | 0°97615
2°86 | 0°94627 3°11 | 0°96406 60 § 3°36 | 0°97657
2°87 | 0°94711 3°12 | 0°96466 58 § 3°37 | 0°97698
2°88 | 0°94793 3°13 | 0°96524 58 § 3°38 | 0:97738
2°89 | 0°94874 3°14 | 0°96582 56 § 3°39 | Q:97778

| 2:90 | 0-94954 3°15 | 0°96638 56 § 3°40 | 0°97817
2°91 | 0°95033 3°16 | 0°96694 55 § 3°50! 0°98176
2:92 | O0°95111 3:17 | 0°96749 55 § 3°60 | 0°98482
52°93 | 0°95187 3°18 | 0°96804 53 § 3°70 | 0°98743
2°94 | 095263 3:19 | 0°96857 53 § 3°80 | 0-98962
2°95 | 0°95338 3°20 |; 0°96910 52 § 3°90 | 0°99147
2°96 | 0°95412 3°21 | 0°96962 51 § 4°00 | 0°99302
2°97 | 0°95485 3°22 | 0°97013 51 § 4:10 | 0°99431
2°98 | 0°95557 3:23 | 0°97064 50 § 4:20 | 0°99639
2°99 | 0°95628 3:24 | 0°97114 49 — 4:30 | 0°99627
3°00 | 0°95698 3°25 | 0°97163 48 — 4°40] 0°99700
3°01 | 0°95767 3:26 | 0°97211 48 § 4°50! 0°9976Q
3°02 , 0°95835 3°27 | 06:97259 47 § 4°60 | 0°99808
3°03 | 0°95902 3°28 | 0°97306 46 § 4°70 | 0:99848
3°04 ;} 0°95968 3°29 | 0°97352 45 § 4:80 | 0°99879
3°05 | 0°96033 3°30 | 0°97397 45 §| 4°90 | 0°99905
3°06 | 0°96098 3°31 | 0°27442 44 § 5:00 | 0°99926
3°07 | 0°96161 3°32 | 0°97486 44
3°08 | 0°96224 3°33 ; 0°97530 43
3°09 | 0°96286 3°34 | 0°97573 42
THE END.
Lonpon:

Ay

Pio

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